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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k0558401.png" /> be a complex linear space on a which a Hermitian sesquilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k0558402.png" /> is defined (i.e. a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k0558403.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k0558404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k0558405.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k0558406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k0558407.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k0558408.png" /> (or, more exactly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k0558409.png" />) is called a Krein space if in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584010.png" /> there are two linear manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584011.png" /> such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584014.png" /> are Hilbert spaces (cf. [[Hilbert space|Hilbert space]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584015.png" />. It is always assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584016.png" /> (otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584017.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584018.png" /> is a Hilbert space); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584019.png" /> is called the indefinite inner product of the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584020.png" />. If, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584022.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584024.png" />-space or Pontryagin space of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584026.png" /> (cf. also [[Pontryagin space|Pontryagin space]]); in the sequel, for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584027.png" />-space it is always assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584028.png" />.
+
Let  $  {\mathcal K} $
 +
be a complex linear space on a which a Hermitian sesquilinear form  $  [ \cdot , \cdot ] $
 +
is defined (i.e. a mapping  $  [ \cdot , \cdot ] :  {\mathcal K} \times {\mathcal K} \rightarrow \mathbf C $
 +
such that  $  [ \alpha _ {1} x _ {1} + \alpha _ {2} x _ {2} , y ] = \alpha _ {1} [ x _ {1} , y ] + \alpha _ {2} [ x _ {2} , y ] $
 +
and $  [ x , y ] = \overline{ {[ y , x ] }}\; $
 +
for all  $  x _ {1} , x _ {2} , x , y \in {\mathcal K} $,
 +
$  \alpha _ {1} , \alpha _ {2} \in \mathbf C $).
 +
Then  $  {\mathcal K} $(
 +
or, more exactly,  $  ( {\mathcal K} , [ \cdot , \cdot ] ) $)
 +
is called a Krein space if in  $  {\mathcal K} $
 +
there are two linear manifolds  $  {\mathcal K} _  \pm  $
 +
such that
 +
 
 +
$$ \tag{a1 }
 +
{\mathcal K}  = {\mathcal K} _ {+} \dot{+} {\mathcal K} _ {-} ,
 +
$$
 +
 
 +
$  ( {\mathcal K} _ {+} , [ \cdot , \cdot ] ) $
 +
and  $  ( {\mathcal K} _ {-} , - [ \cdot , \cdot ] ) $
 +
are Hilbert spaces (cf. [[Hilbert space|Hilbert space]]) and $  [ {\mathcal K} _ {+} , {\mathcal K} _ {-} ] = \{ 0 \} $.  
 +
It is always assumed that $  {\mathcal K} _ {+} , {\mathcal K} _ {-} \neq \{ 0 \} $(
 +
otherwise $  ( {\mathcal K} , [ \cdot , \cdot ] ) $
 +
or $  ( {\mathcal K} , - [ \cdot , \cdot ] ) $
 +
is a Hilbert space); $  [ \cdot , \cdot ] $
 +
is called the indefinite inner product of the Krein space $  {\mathcal K} $.  
 +
If, in particular, $  \kappa = \min (  \mathop{\rm dim}  {\mathcal K} _ {+} ,  \mathop{\rm dim}  {\mathcal K} _ {-} ) < \infty $,  
 +
then $  {\mathcal K} _ {-} $
 +
is a $  \pi _  \kappa  $-
 +
space or Pontryagin space of index $  \kappa $(
 +
cf. also [[Pontryagin space|Pontryagin space]]); in the sequel, for a $  \pi _  \kappa  $-
 +
space it is always assumed that $  \kappa = \mathop{\rm dim}  {\mathcal K} _ {+} $.
  
Using the decomposition (a1), on the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584029.png" /> a Hilbert inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584030.png" /> can be defined as follows:
+
Using the decomposition (a1), on the Krein space $  ( {\mathcal K} , [ \cdot , \cdot ] ) $
 +
a Hilbert inner product $  ( \cdot , \cdot ) $
 +
can be defined as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
( x , y )  = [ x _ {+} , y _ {+} ] - [ x _ {-} , y _ {-} ] ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584032.png" /></td> </tr></table>
+
$$
 +
= x _ {+} + x _ {-} ,\  y  = y _ {+} + y _ {-} ,\  x _  \pm  , y _  \pm  \in  {\mathcal K} _ {+} .
 +
$$
  
Although the decomposition (a1) is not unique, the decompositions of the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584033.png" /> are uniquely determined and the Hilbert norms, generated by different decompositions (a1) according to (a2), are equivalent. All topological notions in a Krein space, if not stated explicitly otherwise, refer to this topology. In the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584034.png" />, the orthogonal projections onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584036.png" /> are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584038.png" />, respectively. Then for the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584039.png" />, called a fundamental symmetry, one has
+
Although the decomposition (a1) is not unique, the decompositions of the components $  {\mathcal K} _  \pm  $
 +
are uniquely determined and the Hilbert norms, generated by different decompositions (a1) according to (a2), are equivalent. All topological notions in a Krein space, if not stated explicitly otherwise, refer to this topology. In the Hilbert space $  ( {\mathcal K} , ( \cdot , \cdot ) ) $,  
 +
the orthogonal projections onto $  {\mathcal K} _ {+} $
 +
and $  {\mathcal K} _ {-} $
 +
are denoted by $  P _ {+} $
 +
and $  P _ {-} $,  
 +
respectively. Then for the operator $  J = P _ {+} - P _ {-} $,  
 +
called a fundamental symmetry, one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
$$ \tag{a3 }
 +
[ x , y ]  = ( J x , y ) ,\ \
 +
x , y \in {\mathcal K} ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584041.png" /> has the properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584043.png" />. Conversely, given a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584044.png" /> and in it an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584045.png" /> with these properties (or, more generally, an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584046.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584048.png" />), then an indefinite inner product is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584049.png" /> by (a3) (or, respectively, by the relation
+
and $  J $
 +
has the properties: $  J  ^ {2} = I $,  
 +
$  J = J  ^ {*} $.  
 +
Conversely, given a Hilbert space $  ( {\mathcal K} , ( \cdot , \cdot ) ) $
 +
and in it an operator $  J $
 +
with these properties (or, more generally, an operator $  G $
 +
with $  G = G  ^ {*} $,  
 +
0 \in \rho ( G) $),  
 +
then an indefinite inner product is defined on $  {\mathcal K} $
 +
by (a3) (or, respectively, by the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
$$ \tag{a4 }
 +
[ x , y ]  = ( G x , y ) ,\ \
 +
x , y \in {\mathcal K} \textrm{ ) } ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584051.png" /> is a Krein space. Because of this construction, Krein spaces are sometimes called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584053.png" />-spaces.
+
and $  ( {\mathcal K} , [ \cdot , \cdot ] ) $
 +
is a Krein space. Because of this construction, Krein spaces are sometimes called $  J $-
 +
spaces.
  
If, more generally, a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584054.png" /> and a bounded self-adjoint, not semi-definite, operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584055.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584056.png" /> are given, the relation (a4) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584057.png" /> defines a Hermitian sesquilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584059.png" />. This form can be extended by continuity to the completion of the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584060.png" /> with respect to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584061.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584062.png" />). This completion, equipped with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584063.png" />, is a Krein space containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584064.png" /> as a dense subset.
+
If, more generally, a Hilbert space $  ( {\mathcal H} , ( \cdot , \cdot ) ) $
 +
and a bounded self-adjoint, not semi-definite, operator $  G $
 +
in $  {\mathcal H} $
 +
are given, the relation (a4) with $  x , y \in {\mathcal H} $
 +
defines a Hermitian sesquilinear form $  [ \cdot , \cdot ] $
 +
on $  {\mathcal H} $.  
 +
This form can be extended by continuity to the completion of the quotient space $  {\mathcal H} / \mathop{\rm Ker}  G $
 +
with respect to the norm $  \| | G |  ^ {1/2} x \| $(
 +
$  x \in {\mathcal H} $).  
 +
This completion, equipped with $  [ \cdot , \cdot ] $,  
 +
is a Krein space containing $  {\mathcal H} / \mathop{\rm Ker}  G $
 +
as a dense subset.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584065.png" /> is a real and locally summable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584066.png" /> which assumes positive and negative values on sets of positive [[Lebesgue measure|Lebesgue measure]], then the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584067.png" /> of all (classes of) measurable functions (cf. [[Measurable function|Measurable function]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584068.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584069.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584070.png" /> and equipped with the indefinite inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584071.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584072.png" />) is a Krein space. More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584073.png" /> is a real function which is locally of bounded variation and not isotone on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584075.png" /> denotes its total variation, then the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584076.png" />, of all measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584077.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584078.png" /> and equipped with the indefinite inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584079.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584080.png" />) is a Krein space.
+
If $  r $
 +
is a real and locally summable function on $  \mathbf R $
 +
which assumes positive and negative values on sets of positive [[Lebesgue measure|Lebesgue measure]], then the space $  L _ {2,r} $
 +
of all (classes of) measurable functions (cf. [[Measurable function|Measurable function]]) $  f $
 +
on $  \mathbf R $
 +
such that $  \int _ {- \infty }  ^  \infty  | f  |  ^ {2} |  r |  d x < \infty $
 +
and equipped with the indefinite inner product $  [ f , g ] = \int _ {- \infty }  ^  \infty  f \overline{g}\; r  d x $(
 +
$  f , g \in L _ {2,r} $)  
 +
is a Krein space. More generally, if $  \sigma $
 +
is a real function which is locally of bounded variation and not isotone on $  \mathbf R $
 +
and $  | \sigma | $
 +
denotes its total variation, then the space $  L _ {2} ( \sigma ) $,  
 +
of all measurable functions $  f $
 +
such that $  \int _ {- \infty }  ^  \infty  | f |  ^ {2}  d | \sigma | < \infty $
 +
and equipped with the indefinite inner product $  [ f , g ] = \int _ {- \infty }  ^ {- \infty } f \overline{g}\;  d \sigma $(
 +
$  f , g \in L _ {2} ( \sigma ) $)  
 +
is a Krein space.
  
Further, a complex linear space with a Hermitian sesquilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584081.png" />, which has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584082.png" /> negative squares (that is, each linear manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584083.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584084.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584086.png" />, is of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584087.png" /> and at least one such manifold is of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584088.png" />), can be canonically imbedded into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584089.png" />-space by taking a quotient space and completing it (see [[#References|[a4]]], [[#References|[a2]]], [[#References|[a9]]], [[#References|[a11]]]).
+
Further, a complex linear space with a Hermitian sesquilinear form $  [ \cdot , \cdot ] $,  
 +
which has $  \kappa $
 +
negative squares (that is, each linear manifold $  {\mathcal L} \subset  {\mathcal K} $
 +
with $  [ x , x ] < 0 $
 +
for $  x \in {\mathcal L} $,  
 +
$  x \neq 0 $,  
 +
is of dimension $  \leq  \kappa $
 +
and at least one such manifold is of dimension $  \kappa $),  
 +
can be canonically imbedded into a $  \pi _  \kappa  $-
 +
space by taking a quotient space and completing it (see [[#References|[a4]]], [[#References|[a2]]], [[#References|[a9]]], [[#References|[a11]]]).
  
The indefinite inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584090.png" /> on the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584091.png" /> gives rise to a classification of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584092.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584093.png" /> is called positive, non-negative, neutral, etc. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584096.png" />, etc. A linear manifold or a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584097.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584098.png" /> is called positive, non-negative, neutral, etc. if all its non-zero elements are positive, non-negative, neutral, etc. The set of all, e.g., non-negative elements is not linear, but it contains subspaces, and among them maximal ones, called maximal non-negative subspaces. All maximal non-negative subspaces of the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k05584099.png" /> have the same dimension (as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840100.png" />). A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840101.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840102.png" /> (with the decomposition (a1)) is maximal non-negative if and only if it can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840103.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840104.png" />, the angular operator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840105.png" />, is a [[Contraction(2)|contraction]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840106.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840107.png" />. A dual pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840108.png" /> of subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840109.png" /> is defined as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840110.png" /> is a non-negative subspace, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840111.png" /> is a non-positive subspace and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840112.png" />. Any dual pairs is contained in a maximal dual pair (maximality of dual pairs is defined in a natural way by inclusion); in a maximal dual pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840113.png" /> the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840114.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840115.png" />) is maximal non-negative (respectively, non-positive) (R.S. Phillips).
+
The indefinite inner product $  [ \cdot , \cdot ] $
 +
on the Krein space $  {\mathcal K} $
 +
gives rise to a classification of the elements of $  {\mathcal K} $:  
 +
$  x \in {\mathcal K} $
 +
is called positive, non-negative, neutral, etc. if $  [ x , x ] > 0 $,
 +
$  [ x , x ] < 0 $,
 +
$  [ x , x ] = 0 $,  
 +
etc. A linear manifold or a subspace $  {\mathcal L} $
 +
in $  {\mathcal K} $
 +
is called positive, non-negative, neutral, etc. if all its non-zero elements are positive, non-negative, neutral, etc. The set of all, e.g., non-negative elements is not linear, but it contains subspaces, and among them maximal ones, called maximal non-negative subspaces. All maximal non-negative subspaces of the Krein space $  {\mathcal K} $
 +
have the same dimension (as $  {\mathcal K} _ {+} $).  
 +
A subspace $  {\mathcal L} $
 +
of $  {\mathcal K} $(
 +
with the decomposition (a1)) is maximal non-negative if and only if it can be written as $  {\mathcal L} = \{ {x _ {+} + K _  {\mathcal L}  x _ {+} } : {x _ {+} \in {\mathcal K} _ {+} } \} $,  
 +
where $  K _  {\mathcal L}  $,  
 +
the angular operator of $  {\mathcal L} $,  
 +
is a [[Contraction(2)|contraction]] from $  ( {\mathcal K} _ {+} , [ \cdot , \cdot ] ) $
 +
into $  ( {\mathcal K} _ {-} , [ \cdot , \cdot ] ) $.  
 +
A dual pair $  ( {\mathcal L} _ {+} , {\mathcal L} _ {-} ) $
 +
of subspaces of $  {\mathcal K} $
 +
is defined as follows: $  {\mathcal L} _ {+} $
 +
is a non-negative subspace, $  {\mathcal L} _ {-} $
 +
is a non-positive subspace and $  [ {\mathcal L} _ {+} , {\mathcal L} _ {-} ] = \{ 0 \} $.  
 +
Any dual pairs is contained in a maximal dual pair (maximality of dual pairs is defined in a natural way by inclusion); in a maximal dual pair $  ( {\mathcal L} _ {+} , {\mathcal L} _ {-} ) $
 +
the subspace $  {\mathcal L} _ {+} $(
 +
respectively, $  {\mathcal L} _ {-} $)  
 +
is maximal non-negative (respectively, non-positive) (R.S. Phillips).
  
Using the indefinite inner product, orthogonality can be defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840116.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840117.png" /> are called orthogonal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840118.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840119.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840120.png" />. Some properties of orthogonality in a Hilbert space are preserved; however, there are also essential differences; e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840121.png" /> can contain non-zero vectors; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840122.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840123.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840124.png" /> is neutral, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840125.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840126.png" />.
+
Using the indefinite inner product, orthogonality can be defined in $  {\mathcal K} $:  
 +
$  x , y \in {\mathcal K} $
 +
are called orthogonal if $  [ x , y ] = 0 $;  
 +
if $  {\mathcal L} \subset  {\mathcal K} $,
 +
then  $  {\mathcal L}  ^  \perp  = \{ {x } : {[ x , {\mathcal L} ] = \{ 0 \} } \} $.  
 +
Some properties of orthogonality in a Hilbert space are preserved; however, there are also essential differences; e.g., $  {\mathcal L} \cap {\mathcal L}  ^  \perp  $
 +
can contain non-zero vectors; $  {\mathcal L} \cap {\mathcal L}  ^  \perp  $
 +
coincides with $  {\mathcal L} $
 +
if $  {\mathcal L} $
 +
is neutral, and $  {\mathcal L} \cap {\mathcal L}  ^  \perp  = \{ 0 \} $
 +
is equivalent to $  \overline{ {{\mathcal L} + {\mathcal L}  ^  \perp  }}\; = {\mathcal K} $.
  
For a densely-defined [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840127.png" /> in the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840128.png" /> an adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840129.png" /> (sometimes called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840131.png" />-adjoint) is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840132.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840134.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840135.png" /> denotes the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840136.png" /> in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840137.png" /> (see (a2)), then evidently <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840138.png" />. Now in the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840139.png" /> classes of operators are defined more or less similarly to the case of a Hilbert space: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840140.png" /> is symmetric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840141.png" />, self-adjoint if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840142.png" />, dissipative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840143.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840144.png" />), contractive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840145.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840146.png" />), unitary if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840147.png" /> is bounded, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840148.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840149.png" />, etc. Also, new classes of operators arise: E.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840150.png" /> is a plus-operator if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840151.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840152.png" />, and a doubly plus-operator if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840153.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840154.png" /> are plus-operators. In a Krein space a densely-defined [[Isometric operator|isometric operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840155.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840156.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840157.png" />) need not be continuous. As in a Hilbert space, self-adjoint and unitary, symmetric and isometric, dissipative and contractive operators are related by the [[Cayley transform|Cayley transform]]. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840159.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840160.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840161.png" /> is unitary.
+
For a densely-defined [[Linear operator|linear operator]] $  T $
 +
in the Krein space $  ( {\mathcal K} , [ \cdot , \cdot ] ) $
 +
an adjoint $  T ^ { + } $(
 +
sometimes called $  J $-
 +
adjoint) is defined by $  [ T x , y ] = [ x , T ^ { + } y ] $(
 +
$  x \in {\mathcal D} ( T) $,  
 +
$  y \in {\mathcal D} ( T ^ { + } ) $).  
 +
If $  T ^ { * } $
 +
denotes the adjoint of $  T $
 +
in the Hilbert space $  ( {\mathcal K} , ( \cdot , \cdot ) ) $(
 +
see (a2)), then evidently $  T ^ { + } = J T ^ { * } J $.  
 +
Now in the Krein space $  {\mathcal K} $
 +
classes of operators are defined more or less similarly to the case of a Hilbert space: $  T $
 +
is symmetric if $  T \subset  T ^ { + } $,  
 +
self-adjoint if $  T = T ^ { + } $,  
 +
dissipative if $  \mathop{\rm Im}  [ T x , x ] \geq  0 $(
 +
$  x \in {\mathcal D} ( T) $),  
 +
contractive if $  [ T x , T x ] \leq  [ x , x ] $(
 +
$  x \in {\mathcal K} $),  
 +
unitary if $  T $
 +
is bounded, $  {\mathcal D} ( T) = {\mathcal K} $
 +
and $  T ^ { + } T = I = T T ^ { + } $,  
 +
etc. Also, new classes of operators arise: E.g., $  T $
 +
is a plus-operator if $  [ x , x ] \geq  0 $
 +
implies $  [ T x , T x ] \geq  0 $,  
 +
and a doubly plus-operator if $  T $
 +
and $  T ^ { + } $
 +
are plus-operators. In a Krein space a densely-defined [[Isometric operator|isometric operator]] $  T $(
 +
i.e. $  [ T x , T y ] = [ x , y ] $
 +
for all $  x , y \in {\mathcal D} ( T) $)  
 +
need not be continuous. As in a Hilbert space, self-adjoint and unitary, symmetric and isometric, dissipative and contractive operators are related by the [[Cayley transform|Cayley transform]]. E.g., if $  A = A  ^ {+} $,
 +
$  z _ {0} \neq \overline{z}\; _ {0} $
 +
and $  z _ {0} \in \rho ( A) $,  
 +
then $  U = ( A - \overline{z}\; _ {0} ) ( A - z _ {0} )  ^ {-} 1 $
 +
is unitary.
  
The spectrum of a self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840162.png" /> in a Krein space is not necessarily real (it can even cover the whole plane), but it is symmetric with respect to the real axis. Similarly, the spectrum of a unitary operator is symmetric with respect to the unit circle.
+
The spectrum of a self-adjoint operator $  A $
 +
in a Krein space is not necessarily real (it can even cover the whole plane), but it is symmetric with respect to the real axis. Similarly, the spectrum of a unitary operator is symmetric with respect to the unit circle.
  
 
The indefinite inner product sometimes gives a classification of the points of the spectrum of an operator: An eigen value is said to be of positive type (negative type, etc.) if the corresponding eigen space is positive (negative, etc.).
 
The indefinite inner product sometimes gives a classification of the points of the spectrum of an operator: An eigen value is said to be of positive type (negative type, etc.) if the corresponding eigen space is positive (negative, etc.).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840163.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840164.png" /> are isolated eigen values of the self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840165.png" /> in a Krein space, then for the corresponding Riesz projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840166.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840167.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840168.png" />, and if, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840169.png" />, then the restrictions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840170.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840171.png" /> have the same Jordan structure. If in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840172.png" />-space the symmetric operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840173.png" /> has a real non-semi-simple eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840174.png" />, then the corresponding algebraic eigen space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840175.png" /> can be decomposed into a direct orthogonal sum: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840176.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840177.png" /> is a positive subspace contained in the geometric eigen space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840178.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840179.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840180.png" /> is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840181.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840182.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840183.png" /> are the lengths of the Jordan chains of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840184.png" />, one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840185.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840186.png" /> is a non-real eigen value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840187.png" />, one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840188.png" /> as the dimension of the corresponding algebraic eigen space. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840189.png" />, where the sum extends over all eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840190.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840191.png" /> in the closed upper half-plane. In particular, the length of any Jordan chain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840192.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840193.png" />, and the number of eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840194.png" /> in the open upper half-plane, and also the number of non-semi-simple eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840195.png" />, does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840196.png" />.
+
If $  \lambda $,  
 +
$  \overline \lambda \; $
 +
are isolated eigen values of the self-adjoint operator $  A $
 +
in a Krein space, then for the corresponding Riesz projections $  E _  \lambda  $,  
 +
$  E _ {\overline \lambda \; }  $
 +
one has $  E _ {\overline \lambda \; }  = E _  \lambda  ^ {+} $,  
 +
and if, e.g., $  \mathop{\rm dim}  {\mathcal R} ( E _  \lambda  ) < \infty $,  
 +
then the restrictions $  A \mid  _ { {\mathcal R} ( E _  \lambda  ) } $
 +
and $  A \mid  _ { {\mathcal R} ( E _ {\overline \lambda \; }  ) } $
 +
have the same Jordan structure. If in a $  \pi _  \kappa  $-
 +
space the symmetric operator $  A $
 +
has a real non-semi-simple eigen value $  \lambda $,  
 +
then the corresponding algebraic eigen space $  {\mathcal E} _  \lambda  $
 +
can be decomposed into a direct orthogonal sum: $  {\mathcal E} _  \lambda  = {\mathcal E} _  \lambda  ^ { \prime } + {\mathcal E} _  \lambda  ^ { \prime\prime } $,  
 +
where $  {\mathcal E} _  \lambda  ^ { \prime\prime } $
 +
is a positive subspace contained in the geometric eigen space of $  A $
 +
at $  \lambda $,  
 +
and $  {\mathcal E} _  \lambda  ^ { \prime } \neq \{ 0 \} $
 +
is invariant under $  A $
 +
with $  \mathop{\rm dim}  {\mathcal E} _  \lambda  ^ { \prime } < \infty $;  
 +
if $  d _ {1} \dots d _ {r} $
 +
are the lengths of the Jordan chains of $  A \mid  _ { {\mathcal E} _  \lambda  ^ { \prime } } $,  
 +
one puts $  \rho ( \lambda )= \sum _ {j=} 1  ^  \kappa  [ d _ {j} /2 ] $;  
 +
if $  \lambda $
 +
is a non-real eigen value of $  A $,  
 +
one defines $  \rho ( \lambda ) $
 +
as the dimension of the corresponding algebraic eigen space. Then $  \sum \rho ( \lambda ) \leq  \kappa $,  
 +
where the sum extends over all eigen values $  \lambda $
 +
of $  A $
 +
in the closed upper half-plane. In particular, the length of any Jordan chain of $  A $
 +
is $  \leq  2 \kappa + 1 $,  
 +
and the number of eigen values of $  A $
 +
in the open upper half-plane, and also the number of non-semi-simple eigen values of $  A $,  
 +
does not exceed $  \kappa $.
  
Specific results for Krein spaces are statements about the existence of maximal non-negative (or maximal non-positive) subspaces, which are invariant under a given operator. The first general result of this type was proved by L.S. Pontryagin in 1944, stating that a self-adjoint operator in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840197.png" />-space has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840198.png" />-dimensional non-positive (that is, a maximal non-positive) invariant subspace. Subsequently, similar results were proved for various classes of operators in Krein spaces. E.g., a bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840199.png" /> in a Krein space has a maximal non-negative invariant subspace if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840200.png" /> is compact and, additionally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840201.png" /> is self-adjoint or dissipative or unitary or a plus-operator, etc. (see [[#References|[a2]]], [[#References|[a4]]]). One possibility for proving these results, e.g. for a unitary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840202.png" />, is to establish the existence of a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840203.png" /> of the fractional-linear transformation
+
Specific results for Krein spaces are statements about the existence of maximal non-negative (or maximal non-positive) subspaces, which are invariant under a given operator. The first general result of this type was proved by L.S. Pontryagin in 1944, stating that a self-adjoint operator in a $  \pi _  \kappa  $-
 +
space has a $  \kappa $-
 +
dimensional non-positive (that is, a maximal non-positive) invariant subspace. Subsequently, similar results were proved for various classes of operators in Krein spaces. E.g., a bounded linear operator $  T $
 +
in a Krein space has a maximal non-negative invariant subspace if $  P _ {+} T P _ {-} $
 +
is compact and, additionally, $  T $
 +
is self-adjoint or dissipative or unitary or a plus-operator, etc. (see [[#References|[a2]]], [[#References|[a4]]]). One possibility for proving these results, e.g. for a unitary operator $  T $,  
 +
is to establish the existence of a fixed point $  K _ {0} $
 +
of the fractional-linear transformation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840204.png" /></td> </tr></table>
+
$$
 +
K  \rightarrow  ( T _ {21} + T _ {22} K ) ( T _ {11} + T _ {12} K )  ^ {-} 1 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840205.png" /> is a contraction from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840206.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840207.png" /> (an angular operator) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840208.png" /> is the matrix representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840209.png" /> with respect to (a1). By different methods also in other cases the existence of a maximal non-negative invariant subspace has been proved, e.g.: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840210.png" /> is unitary and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840211.png" /> is uniformly bounded for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840212.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840213.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840214.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840215.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840216.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840217.png" /> is bounded, self-adjoint and there exists a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840218.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840219.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840220.png" />). In many cases these maximal non-positive invariant subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840221.png" /> can be specified by properties of the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840222.png" />. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840223.png" /> is bounded, self-adjoint and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840224.png" /> is compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840225.png" /> can be chosen such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840226.png" />. There are also results about the existence of a common invariant maximal non-positive subspace for a commuting family of operators, e.g.: A commuting family of bounded self-adjoint operators in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840227.png" />-space has a common maximal non-negative invariant subspace (M.A. Naimark; for applications in the representation theory of groups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840228.png" />-spaces see [[#References|[a19]]]). Phillips asked ([[#References|[a16]]]) if a dual pair of subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840229.png" /> which are invariant under a commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840230.png" /> of bounded self-adjoint operators in the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840231.png" /> can always be extended to a maximal dual pair whose subspaces are still invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840232.png" /> (which would imply that each bounded self-adjoint operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840233.png" /> has a maximal non-negative invariant subspace). Only partial solutions to this problem are known (cf. [[#References|[a4]]], [[#References|[a2]]], [[#References|[a14]]]).
+
where $  K $
 +
is a contraction from $  {\mathcal K} _ {+} $
 +
into $  {\mathcal K} _ {-} $(
 +
an angular operator) and $  ( T _ {ij} ) _ {1}  ^ {2} $
 +
is the matrix representation of $  T $
 +
with respect to (a1). By different methods also in other cases the existence of a maximal non-negative invariant subspace has been proved, e.g.: 1) $  T $
 +
is unitary and $  \| T ^ { n } \| $
 +
is uniformly bounded for all $  n = 0 , 1 ,\dots $;  
 +
2) $  [ T x , T x ] > [ x , x ] $
 +
for all $  x \in {\mathcal K} $,  
 +
$  x \neq 0 $,  
 +
and $  \sigma ( T) \cap \{ | \rho | = 1 \} = \emptyset $;  
 +
and 3) $  T $
 +
is bounded, self-adjoint and there exists a polynomial $  p $
 +
such that $  [ p ( T) x , x ] \geq  0 $(
 +
$  x \in {\mathcal K} $).  
 +
In many cases these maximal non-positive invariant subspaces $  {\mathcal L} $
 +
can be specified by properties of the spectrum of $  A \mid  _  {\mathcal L}  $.  
 +
E.g., if $  T $
 +
is bounded, self-adjoint and $  P _ {+} T P _ {-} $
 +
is compact, then $  {\mathcal L} $
 +
can be chosen such that $  \mathop{\rm Im}  \sigma ( A \mid  _  {\mathcal L}  ) \geq  0 $.  
 +
There are also results about the existence of a common invariant maximal non-positive subspace for a commuting family of operators, e.g.: A commuting family of bounded self-adjoint operators in a $  \pi _  \kappa  $-
 +
space has a common maximal non-negative invariant subspace (M.A. Naimark; for applications in the representation theory of groups in $  \pi _  \kappa  $-
 +
spaces see [[#References|[a19]]]). Phillips asked ([[#References|[a16]]]) if a dual pair of subspaces of $  {\mathcal K} $
 +
which are invariant under a commutative algebra $  A $
 +
of bounded self-adjoint operators in the Krein space $  {\mathcal K} $
 +
can always be extended to a maximal dual pair whose subspaces are still invariant under $  A $(
 +
which would imply that each bounded self-adjoint operator in $  {\mathcal K} $
 +
has a maximal non-negative invariant subspace). Only partial solutions to this problem are known (cf. [[#References|[a4]]], [[#References|[a2]]], [[#References|[a14]]]).
  
A self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840234.png" /> in the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840235.png" /> is called definitizable (positizable in [[#References|[a4]]]) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840236.png" /> and if there exists a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840237.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840238.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840239.png" />). Each self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840240.png" /> in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840241.png" />-space has this property (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840242.png" /> can be chosen to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840243.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840244.png" /> the minimal polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840245.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840246.png" /> being a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840247.png" />-dimensional non-positive invariant subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840248.png" />); also, each self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840249.png" /> in a Krein space for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840250.png" /> and for which the Hermitian sesquilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840251.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840252.png" />) has a finite number of negative squares, is definitizable.
+
A self-adjoint operator $  A $
 +
in the Krein space $  {\mathcal K} $
 +
is called definitizable (positizable in [[#References|[a4]]]) if $  \rho ( A) \neq \emptyset $
 +
and if there exists a polynomial $  p $
 +
such that $  [ p ( A) x , x ] \geq  0 $(
 +
$  x \in {\mathcal D} ( p ( A) ) $).  
 +
Each self-adjoint operator $  A $
 +
in a $  \pi _  \kappa  $-
 +
space has this property (where $  p $
 +
can be chosen to be $  q \overline{q}\; $
 +
with $  q $
 +
the minimal polynomial of $  A \mid  _  {\mathcal L}  $,  
 +
$  {\mathcal L} $
 +
being a $  \kappa $-
 +
dimensional non-positive invariant subspace of $  A $);  
 +
also, each self-adjoint operator $  A $
 +
in a Krein space for which $  \rho ( A ) \neq \emptyset $
 +
and for which the Hermitian sesquilinear form $  [ A x , y ] $(
 +
$  x , y \in {\mathcal D} ( A) $)  
 +
has a finite number of negative squares, is definitizable.
  
The non-real spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840253.png" /> of the definitizable operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840254.png" /> consists of at most finitely many eigen values, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840255.png" /> has a [[Spectral function|spectral function]], with possibly certain critical points [[#References|[a13]]], [[#References|[a2]]]. This means that there is a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840256.png" /> (of critical points) such that on the semi-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840257.png" />, consisting of all bounded intervals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840258.png" /> with end points not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840259.png" /> and their complements, a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840260.png" /> with values in the set of all self-adjoint projections in the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840261.png" /> is defined, such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840262.png" />: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840263.png" /> is a positive (negative) subspace if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840264.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840265.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840266.png" /> for some definitizing polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840267.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840268.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840269.png" /> is in the double commutant of the [[Resolvent|resolvent]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840270.png" />; and c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840271.png" /> is bounded, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840272.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840273.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840274.png" />. If, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840275.png" /> is bounded and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840276.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840277.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840278.png" />, and one has
+
The non-real spectrum $  \sigma _ {0} ( A) $
 +
of the definitizable operator $  A $
 +
consists of at most finitely many eigen values, and $  A $
 +
has a [[Spectral function|spectral function]], with possibly certain critical points [[#References|[a13]]], [[#References|[a2]]]. This means that there is a finite set $  c ( A) \subset  \mathbf R \cup \{ \infty \} $(
 +
of critical points) such that on the semi-ring $  \mathbf R _ {A} $,  
 +
consisting of all bounded intervals of $  \mathbf R $
 +
with end points not in $  c ( A) $
 +
and their complements, a homomorphism $  E $
 +
with values in the set of all self-adjoint projections in the Krein space $  {\mathcal K} $
 +
is defined, such that for $  \Delta \in \mathbf R _ {A} $:  
 +
a) $  E ( \Delta ) {\mathcal K} $
 +
is a positive (negative) subspace if $  p > 0 $(
 +
respectively, $  p < 0 $)  
 +
on $  \overline \Delta \; \cap \sigma ( A) $
 +
for some definitizing polynomial $  p $
 +
of $  A $;  
 +
b) $  E ( \Delta ) $
 +
is in the double commutant of the [[Resolvent|resolvent]] of $  A $;  
 +
and c) if $  \Delta $
 +
is bounded, then $  E ( \Delta ) {\mathcal K} \subset  {\mathcal D} ( A) $
 +
and $  \sigma ( A \mid  _ {E ( \Delta ) {\mathcal K} }  ) \subset  \overline \Delta \; $,  
 +
$  \sigma ( A \mid  _ {( I - E ( \Delta ) ) {\mathcal K} }  ) \subset  \overline{ {( \mathbf R \setminus  \Delta ) }}\; \cup \sigma _ {0} ( A) $.  
 +
If, in particular, $  A $
 +
is bounded and $  [ A x , x ] \geq  0 $(
 +
$  x \in {\mathcal K} $),  
 +
then $  c ( A) \subset  \{ 0 \} $,  
 +
and one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840279.png" /></td> </tr></table>
+
$$
 +
A x  = \int\limits _ {- \| A \| } ^ { {\| }  A \| } \lambda E ( d \lambda ) x + N x ,
 +
$$
  
for some bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840280.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840281.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840282.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840283.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840284.png" />).
+
for some bounded operator $  N $
 +
such that $  N = N  ^ {+} $,
 +
$  N  ^ {2} = 0 $,  
 +
$  [ N x , x ] \geq  0 $(
 +
$  x \in {\mathcal K} $).
  
If the spectrum of a definitizable operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840285.png" /> is discrete, then the linear span of its algebraic eigen spaces is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840286.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840287.png" /> is compact and self-adjoint in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840288.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840289.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840290.png" />, then there is a [[Riesz basis|Riesz basis]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840291.png" /> consisting of eigen and associated vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840292.png" /> (I.S. Iokhvidov).
+
If the spectrum of a definitizable operator $  A $
 +
is discrete, then the linear span of its algebraic eigen spaces is dense in $  {\mathcal K} $;  
 +
if $  A $
 +
is compact and self-adjoint in a $  \pi _  \kappa  $-
 +
space $  {\mathcal K} $
 +
and 0 \notin \sigma _ {p} ( A) $,  
 +
then there is a [[Riesz basis|Riesz basis]] of $  {\mathcal K} $
 +
consisting of eigen and associated vectors of $  A $(
 +
I.S. Iokhvidov).
  
There is a theory of extensions of symmetric operators to self-adjoint operators and of generalized resolvents in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840293.png" />-spaces, and also in Krein spaces, which is similar to the Hilbert space situation. The same is true for dilation theory: Each bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840294.png" /> in a Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840295.png" /> has a unitary dilation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840296.png" /> in some Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840297.png" /> ([[#References|[a2]]]). In this context one has the following result: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840298.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840299.png" /> be Krein spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840300.png" /> a simply-connected open domain with smooth boundary such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840302.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840303.png" /> be a function which is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840304.png" /> whose values are bounded linear operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840305.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840306.png" />. Then there exists a Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840307.png" /> and a unitary operator
+
There is a theory of extensions of symmetric operators to self-adjoint operators and of generalized resolvents in $  \pi _  \kappa  $-
 +
spaces, and also in Krein spaces, which is similar to the Hilbert space situation. The same is true for dilation theory: Each bounded linear operator $  T $
 +
in a Krein space $  {\mathcal K} $
 +
has a unitary dilation $  T $
 +
in some Krein space $  {\mathcal K}  tilde \supset {\mathcal K} $([[#References|[a2]]]). In this context one has the following result: Let $  {\mathcal K} _ {1} $,  
 +
$  {\mathcal K} _ {2} $
 +
be Krein spaces, $  {\mathcal D} $
 +
a simply-connected open domain with smooth boundary such that 0 \in {\mathcal D} $,
 +
$  \overline{ {\mathcal D} }\; \subset  \{ {z } : {| z | < 1 } \} $,  
 +
and let $  \Theta $
 +
be a function which is holomorphic in $  {\mathcal D} $
 +
whose values are bounded linear operators from $  {\mathcal K} _ {1} $
 +
to $  {\mathcal K} _ {2} $.  
 +
Then there exists a Krein space $  {\mathcal K} $
 +
and a unitary operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840308.png" /></td> </tr></table>
+
$$
 +
= \left (
 +
\begin{array}{ll}
 +
U _ {11}  &U _ {12}  \\
 +
U _ {21}  &U _ {22}  \\
 +
\end{array}
 +
\right ) : {\mathcal K} \oplus {\mathcal K} _ {1}  \rightarrow  {\mathcal K} \oplus {\mathcal K} _ {2} ,
 +
$$
  
 
such that
 
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840309.png" /></td> </tr></table>
+
$$
 +
\Theta ( z)  = U _ {22} + z U _ {21} ( I - z U _ {11} )  ^ {-} 1 U _ {12} \ \
 +
( z \in {\mathcal D} )
 +
$$
 +
 
 +
(T.Ya. Azizov, see [[#References|[a2]]], [[#References|[a6]]]; here unitary means that  $  U $
 +
maps the Krein space  $  {\mathcal K} \oplus {\mathcal K} _ {1} $
 +
continuously onto the Krein space  $  {\mathcal K} \oplus {\mathcal K} _ {2} $,
 +
preserving the indefinite inner product).
  
(T.Ya. Azizov, see [[#References|[a2]]], [[#References|[a6]]]; here unitary means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840310.png" /> maps the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840311.png" /> continuously onto the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840312.png" />, preserving the indefinite inner product).
+
Some of the first papers about Krein spaces or, more generally, spaces with indefinite inner product, were stimulated by problems of (quantum) mechanics ([[#References|[a4]]], [[#References|[a2]]]; see also [[#References|[a18]]], [[#References|[a17]]]). Operators in Krein spaces arise also in a natural way in problems in mathematical analysis. Some examples of these are: I) Consider the canonical system of differential equations  $  J \dot{x} ( t) = i H ( t) x ( t) $
 +
on  $  [ 0 , \infty ) $,
 +
where  $  H ( t) $,
 +
$  J $
 +
are  $  ( n \times n ) $
 +
matrices,  $  H ( t ) \geq  0 $,
 +
$  J = J  ^ {*} = J  ^ {-} 1 $,
 +
and let  $  U ( t) $
 +
be the corresponding matrizant (cf. [[Cauchy operator|Cauchy operator]]):  $  J \dot{U} ( t) = i H ( t) U ( t) $,
 +
$  U ( 0) = I _ {n} $.
 +
Then  $  U ( t) $
 +
is  $  J $-
 +
unitary (that is, unitary with respect to the inner product defined in  $  \mathbf C  ^ {n} $
 +
by the matrix  $  J $,
 +
see (a3)), and, e.g., in the stability theory for periodic equations  $  ( H ( t) = H ( T + t ) ) $
 +
the classification of the eigen values of  $  U ( T) $
 +
into those of positive or negative type plays an essential role ([[#References|[a5]]], [[#References|[a8]]]). II) The integral operator  $  x ( \cdot ) \rightarrow \int _ {a}  ^ {b} K ( \cdot , s ) x ( s )  d \sigma ( s) $,
 +
$  \sigma $
 +
real and of bounded variation on the interval  $  [ a , b ] $,
 +
$  K ( s , t ) = \overline{ {K ( t , s ) }}\; $(
 +
$  s , t \in [ a , b ] $),
 +
is self-adjoint in the Krein space $  L _ {2} ( \sigma ) $.  
 +
III) The theory of dual pairs of subspaces of a Krein space and their extensions to maximal dual pairs is related to certain questions in the theory of extensions of dissipative operators in a Hilbert space to maximal dissipative ones. Phillips started these investigations in connection with the Cauchy problem for dissipative hyperbolic and parabolic systems (see [[#References|[a2]]], [[#References|[a4]]] for references). IV) With the monic operator polynomial  $  L ( \lambda ) = \lambda  ^ {n} I + \lambda  ^ {n-} 1 B _ {n-} 1 + \dots + \lambda B _ {1} + B _ {0} $,
 +
$  B _ {j} $
 +
bounded self-adjoint operators in some Hilbert space  $  {\mathcal H} $,  
 +
one can associate the so-called companion operator
  
Some of the first papers about Krein spaces or, more generally, spaces with indefinite inner product, were stimulated by problems of (quantum) mechanics ([[#References|[a4]]], [[#References|[a2]]]; see also [[#References|[a18]]], [[#References|[a17]]]). Operators in Krein spaces arise also in a natural way in problems in mathematical analysis. Some examples of these are: I) Consider the canonical system of differential equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840313.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840314.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840315.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840316.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840317.png" /> matrices, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840318.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840319.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840320.png" /> be the corresponding matrizant (cf. [[Cauchy operator|Cauchy operator]]): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840321.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840322.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840323.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840324.png" />-unitary (that is, unitary with respect to the inner product defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840325.png" /> by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840326.png" />, see (a3)), and, e.g., in the stability theory for periodic equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840327.png" /> the classification of the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840328.png" /> into those of positive or negative type plays an essential role ([[#References|[a5]]], [[#References|[a8]]]). II) The integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840329.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840330.png" /> real and of bounded variation on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840331.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840332.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840333.png" />), is self-adjoint in the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840334.png" />. III) The theory of dual pairs of subspaces of a Krein space and their extensions to maximal dual pairs is related to certain questions in the theory of extensions of dissipative operators in a Hilbert space to maximal dissipative ones. Phillips started these investigations in connection with the Cauchy problem for dissipative hyperbolic and parabolic systems (see [[#References|[a2]]], [[#References|[a4]]] for references). IV) With the monic operator polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840335.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840336.png" /> bounded self-adjoint operators in some Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840337.png" />, one can associate the so-called companion operator
+
$$
 +
= \left (  
 +
\begin{array}{ccccc}
 +
- B _ {n-} 1  &- B _ {n-} 2  &\dots  &- B _ {1}  &- B _ {0}  \\
 +
I  & 0 &\dots  & 0 & 0 \\
 +
\cdot  &\cdot  &\dots  &\cdot  &\cdot  \\
 +
0 & 0 &\dots  & I  & 0 \\
 +
\end{array}
 +
\right ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840338.png" /></td> </tr></table>
+
which is self-adjoint in the Krein space  $  {\mathcal K} = {\mathcal H}  ^ {n} $,
 +
$  [ x , y ] = ( G x , y ) $(
 +
$  x , y \in {\mathcal H}  ^ {n} $),
 +
where  $  ( \cdot , \cdot ) $
 +
is the inner product in  $  {\mathcal H}  ^ {n} $
 +
and
  
which is self-adjoint in the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840339.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840340.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840341.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840342.png" /> is the inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840343.png" /> and
+
$$
 +
= \left (
 +
\begin{array}{cccc}
 +
0  & 0 &\dots  & I  \\
 +
0 & 0 &{}  &B _ {n-} 1  \\
 +
\cdot  &\cdot  &{}  &\cdot  \\
 +
\cdot  &\cdot  &{}  &\cdot  \\
 +
\cdot  &\cdot  &{}  &\cdot  \\
 +
I  &B _ {n-} 1  &\dots  &B _ {1}  \\
 +
\end{array}
 +
\right ) .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840344.png" /></td> </tr></table>
+
If, e.g.,  $  n = 2 $
 +
and  $  B _ {0} $
 +
is compact and  $  \geq  0 $,
 +
the results about the existence of maximal non-negative invariant subspaces mentioned above imply that there exists a bounded linear operator  $  Z $
 +
in  $  {\mathcal H} $
 +
satisfying  $  Z  ^ {2} + B _ {1} Z + B _ {0} = 0 $,
 +
$  Z  ^ {*} Z \leq  B _ {0} $
 +
and  $  \mathop{\rm Im}  \sigma ( Z) \geq  0 $[[#References|[a12]]]. In a similar way, if  $  B \geq  0 $,
 +
$  C = C  ^ {*} $
 +
and  $  A $
 +
are  $  ( n \times n ) $
 +
matrices such that  $  G = ( _ { A }  ^ {-} C  {} _ {B} ^ {A  ^ {*} } ) $
 +
has  $  n $
 +
positive and  $  n $
 +
negative eigen values, the solutions  $  X $
 +
of the matrix [[Riccati equation|Riccati equation]]
  
If, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840345.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840346.png" /> is compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840347.png" />, the results about the existence of maximal non-negative invariant subspaces mentioned above imply that there exists a bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840348.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840349.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840350.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840351.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840352.png" /> [[#References|[a12]]]. In a similar way, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840353.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840354.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840355.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840356.png" /> matrices such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840357.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840358.png" /> positive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840359.png" /> negative eigen values, the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840360.png" /> of the matrix [[Riccati equation|Riccati equation]]
+
$$
 +
X B X + X A + A  ^ {*} X - = 0
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840361.png" /></td> </tr></table>
+
with  $  ( X  ^ {*} - X ) ( A + B X ) \geq  0 $
 +
are in bijective correspondence with all maximal non-positive subspaces which are invariant under the self-adjoint operator  $  T = i ( _ {C}  ^ {A}  {} _ {- A  ^ {*}  } ^ { B } ) $
 +
in the  $  2n $-
 +
dimensional Krein space  $  {\mathcal K} = \mathbf C  ^ {2n} $,
 +
equipped with the indefinite inner product (a4) (see [[#References|[a8]]]). V) If  $  L $
 +
is a formally-symmetric regular ordinary differential operator on the interval  $  [ a , b ] $
 +
with symmetric boundary conditions at  $  a $
 +
and  $  b $,
 +
and  $  r $
 +
is a summable function on  $  [ a , b ] $
 +
which is not of constant sign (a.e.) on  $  [ a , b ] $,
 +
then the differential equation  $  L y - \lambda r y = r f $
 +
leads to a self-adjoint operator  $  A $
 +
in the Krein space  $  {\mathcal K} = L _ {2,r} $
 +
with inner product  $  [ f , g ] = \int _ {a}  ^ {b} f \overline{g}\; r  d x $.
 +
If  $  L $
 +
is semi-bounded from below, the operator  $  A $
 +
is definitizable. VI) Krein spaces can be associated with certain eigen value problems for ordinary differential operators containing the eigen value parameters in the boundary conditions. E.g., consider in  $  L _ {2} = L _ {2} [ 0 , \infty ) $
 +
the problem
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840362.png" /> are in bijective correspondence with all maximal non-positive subspaces which are invariant under the self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840363.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840364.png" />-dimensional Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840365.png" />, equipped with the indefinite inner product (a4) (see [[#References|[a8]]]). V) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840366.png" /> is a formally-symmetric regular ordinary differential operator on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840367.png" /> with symmetric boundary conditions at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840368.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840369.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840370.png" /> is a summable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840371.png" /> which is not of constant sign (a.e.) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840372.png" />, then the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840373.png" /> leads to a self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840374.png" /> in the Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840375.png" /> with inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840376.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840377.png" /> is semi-bounded from below, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840378.png" /> is definitizable. VI) Krein spaces can be associated with certain eigen value problems for ordinary differential operators containing the eigen value parameters in the boundary conditions. E.g., consider in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840379.png" /> the problem
+
$$
 +
-  
 +
\frac{d  ^ {2} y }{d x  ^ {2} }
 +
+
 +
q y - \lambda y  = f ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840380.png" /></td> </tr></table>
+
which is supposed to have a limit point at  $  \infty $
 +
and with a boundary condition  $  \alpha ( \lambda ) y ( 0) + \beta ( \lambda ) y  ^  \prime  ( 0) = 0 $
 +
at  $  x = 0 $(
 +
$  \alpha $,
 +
$  \beta $
 +
are functions which are holomorphic on some set  $  D _ {\alpha , \beta }  \subset  \mathbf C $
 +
and satisfying a symmetry condition). The solution of this problem can be represented as  $  y = P ( A - \lambda I )  ^ {-} 1 f $(
 +
$  f \in L _ {2} $),
 +
where, in general,  $  A $
 +
is a self-adjoint operator in some Krein space  $  {\mathcal K} = L _ {2} \oplus {\mathcal K} _ {1} $
 +
and  $  P $
 +
is the orthogonal projection from  $  {\mathcal K} $
 +
onto  $  L _ {2} $[[#References|[a17]]]. VII) Certain classes of analytic functions are closely related to the theory of operators in  $  \pi _  \kappa  $-
 +
spaces. This concerns, e.g., functions  $  f $
 +
which are defined and meromorphic in the upper half-plane (or the unit disc) and which are such that the kernel
  
which is supposed to have a limit point at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840381.png" /> and with a boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840382.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840383.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840384.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840385.png" /> are functions which are holomorphic on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840386.png" /> and satisfying a symmetry condition). The solution of this problem can be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840387.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840388.png" />), where, in general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840389.png" /> is a self-adjoint operator in some Krein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840390.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840391.png" /> is the orthogonal projection from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840392.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840393.png" /> [[#References|[a17]]]. VII) Certain classes of analytic functions are closely related to the theory of operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840394.png" />-spaces. This concerns, e.g., functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840395.png" /> which are defined and meromorphic in the upper half-plane (or the unit disc) and which are such that the kernel
+
$$
 +
N _ {f} ( z , \rho ) = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840396.png" /></td> </tr></table>
+
\frac{f ( z) - \overline{ {f ( \rho ) }}\; }{z - \overline \rho \; }
 +
 
 +
$$
  
 
(or
 
(or
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840397.png" /></td> </tr></table>
+
$$
 +
\left . S _ {f} ( z , \overline \rho \; = \
 +
 
 +
\frac{1 - f ( z) \overline{ {f ( \rho ) }}\; }{1 - z \overline \rho \; }
 +
\right )
 +
$$
  
has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840398.png" /> negative squares (that is, for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840399.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840400.png" />, the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840401.png" /> has at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840402.png" /> negative eigen values and for at least one choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840403.png" /> it has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840404.png" /> negative eigen values). Corresponding extrapolation or moment problems can be treated by making use of results of the theory of symmetric or isometric operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840405.png" />-spaces (see [[#References|[a12]]], [[#References|[a2]]]).
+
has $  \kappa $
 +
negative squares (that is, for arbitrary $  n $
 +
and $  z _ {1} \dots z _ {n} $,  
 +
the matrix $  ( N _ {f} ( z _ {i} , z _ {j} ) ) _ {1}  ^ {n} $
 +
has at most $  n $
 +
negative eigen values and for at least one choice of $  n , z _ {1} \dots z _ {n} $
 +
it has $  \kappa $
 +
negative eigen values). Corresponding extrapolation or moment problems can be treated by making use of results of the theory of symmetric or isometric operators in $  \pi _  \kappa  $-
 +
spaces (see [[#References|[a12]]], [[#References|[a2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.Ya Azizov,  I.S. Iokhvidov,  "Linear operators in spaces with indefinite metric and their applications"  ''Russian Math. Surveys'' , '''15'''  (1981)  pp. 438–490  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''17'''  (1979)  pp. 113–205</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.Ya Azizov,  I.S. Iokhvidov,  "Foundations of the theory of linear operators in spaces with indefinite metric" , Moscow  (1986)  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Ando,  "Linear operators in Krein spaces" , Hokkaido Univ.  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Bognár,  "Indefinite inner product spaces" , Springer  (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Yu.L. Daletskii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Dijksma,  H. Langer,  H.S.V. de Snoo,  "Unitary colligations in Krein spaces and their role in the extension theory of isometries and symmetric linear relations in Hilbert spaces"  S. Kurepa (ed.)  et al. (ed.) , ''Foundational analysis II'' , ''Lect. notes in math.'' , '''1247''' , Springer  (1987)  pp. 1–42</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Dijksma,  H. Langer,  H.S.V. de Snoo,  "Symmetric Sturm–Liouville operators with eigenvalues depending boundary conditions" , ''Oscillation, Bifurcations and Chaos'' , ''CMS Conf. Proc.'' , '''8''' , Amer. Math. Soc.  (1987)  pp. 87–116</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  I. [I. Gokhberg] Gohberg,  P. Lancaster,  L. Rodman,  "Matrices and indefinite scalar products" , Birkhäuser  (1983)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  I.S. [I.S. Iokhvidov] Iohidov,  M.G. Krein,  H. Langer,  "Introduction to the spectral theory of operators in spaces with an indefinite metric" , Akademie Verlag  (1982)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  V.I. Istraţescu,  "Inner product spaces. Theory and applications" , Reidel  (1987)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  M.G. Krein,  "Introduction to the geometry of indefinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840406.png" />-spaces and the theory of operators in these spaces" , ''Second Math. Summer School'' , '''1''' , Kiev  (1965)  pp. 15–92  (In Russian)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M.G. Krein,  H. Langer,  "Ueber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840407.png" /> zusammenhängen, I: Einige Funktionenklassen und ihre Darstellungen"  ''Math. Nachr.'' , '''77'''  (1977)  pp. 187–236</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  H. Langer,  "Spectral functions of definitizable operators in Krein spaces"  D. Butković (ed.)  et al. (ed.) , ''Functional analysis'' , ''Lect. notes in math.'' , '''948''' , Springer  (1982)  pp. 1–46</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  H. Langer,  "Invariante Teilräume definisierbarer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840408.png" />-selbstadjungierter Operatoren"  ''Ann. Acad. Sci. Fenn A. I'' , '''475'''  (1971)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  J. Milnor,  D. Husemoller,  "Symmetric bilinear forms" , Springer  (1973)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  R.S. Phillips,  "The extensions of dual subspaces invariant under an algebra" , ''Proc. Internat. Symp. Linear Spaces (Jerusalem, 1960)'' , Pergamon  (1961)  pp. 366–398</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  L. Bracci,  G. Morchio,  F. Strocchi,  "Wigner's theorem on symmetries in indefinite metric spaces"  ''Comm. Math. Phys.'' , '''41'''  (1975)  pp. 289–299</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  K.L. Nagy,  "State vector spaces with indefinite metric in quantum field theory" , Noordhoff  (1966)</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  M.A. Naimark,  R.S. Ismagilov,  "Representations of groups and algebras in a space with indefinite metric"  ''Itogi Nauk. i Tekhn. Mat. Anal.''  (1969)  pp. 73–105  (In Russian)</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  M.G. Krein,  H. Langer,  "On some mathematical principles in the linear theory of damped oscillations of continua"  ''Integral Equations, Operator Theory'' , '''1'''  (1978)  pp. 364–399; 539–566  ''Proc. Internat. Symp. Appl. Theory of Functions in Continuum Mechanics, Tbilizi'' , '''2'''  (1963)  pp. 283–322</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.Ya Azizov,  I.S. Iokhvidov,  "Linear operators in spaces with indefinite metric and their applications"  ''Russian Math. Surveys'' , '''15'''  (1981)  pp. 438–490  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''17'''  (1979)  pp. 113–205</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.Ya Azizov,  I.S. Iokhvidov,  "Foundations of the theory of linear operators in spaces with indefinite metric" , Moscow  (1986)  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Ando,  "Linear operators in Krein spaces" , Hokkaido Univ.  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Bognár,  "Indefinite inner product spaces" , Springer  (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Yu.L. Daletskii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Dijksma,  H. Langer,  H.S.V. de Snoo,  "Unitary colligations in Krein spaces and their role in the extension theory of isometries and symmetric linear relations in Hilbert spaces"  S. Kurepa (ed.)  et al. (ed.) , ''Foundational analysis II'' , ''Lect. notes in math.'' , '''1247''' , Springer  (1987)  pp. 1–42</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Dijksma,  H. Langer,  H.S.V. de Snoo,  "Symmetric Sturm–Liouville operators with eigenvalues depending boundary conditions" , ''Oscillation, Bifurcations and Chaos'' , ''CMS Conf. Proc.'' , '''8''' , Amer. Math. Soc.  (1987)  pp. 87–116</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  I. [I. Gokhberg] Gohberg,  P. Lancaster,  L. Rodman,  "Matrices and indefinite scalar products" , Birkhäuser  (1983)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  I.S. [I.S. Iokhvidov] Iohidov,  M.G. Krein,  H. Langer,  "Introduction to the spectral theory of operators in spaces with an indefinite metric" , Akademie Verlag  (1982)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  V.I. Istraţescu,  "Inner product spaces. Theory and applications" , Reidel  (1987)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  M.G. Krein,  "Introduction to the geometry of indefinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840406.png" />-spaces and the theory of operators in these spaces" , ''Second Math. Summer School'' , '''1''' , Kiev  (1965)  pp. 15–92  (In Russian)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M.G. Krein,  H. Langer,  "Ueber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840407.png" /> zusammenhängen, I: Einige Funktionenklassen und ihre Darstellungen"  ''Math. Nachr.'' , '''77'''  (1977)  pp. 187–236</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  H. Langer,  "Spectral functions of definitizable operators in Krein spaces"  D. Butković (ed.)  et al. (ed.) , ''Functional analysis'' , ''Lect. notes in math.'' , '''948''' , Springer  (1982)  pp. 1–46</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  H. Langer,  "Invariante Teilräume definisierbarer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055840/k055840408.png" />-selbstadjungierter Operatoren"  ''Ann. Acad. Sci. Fenn A. I'' , '''475'''  (1971)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  J. Milnor,  D. Husemoller,  "Symmetric bilinear forms" , Springer  (1973)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  R.S. Phillips,  "The extensions of dual subspaces invariant under an algebra" , ''Proc. Internat. Symp. Linear Spaces (Jerusalem, 1960)'' , Pergamon  (1961)  pp. 366–398</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  L. Bracci,  G. Morchio,  F. Strocchi,  "Wigner's theorem on symmetries in indefinite metric spaces"  ''Comm. Math. Phys.'' , '''41'''  (1975)  pp. 289–299</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  K.L. Nagy,  "State vector spaces with indefinite metric in quantum field theory" , Noordhoff  (1966)</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  M.A. Naimark,  R.S. Ismagilov,  "Representations of groups and algebras in a space with indefinite metric"  ''Itogi Nauk. i Tekhn. Mat. Anal.''  (1969)  pp. 73–105  (In Russian)</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  M.G. Krein,  H. Langer,  "On some mathematical principles in the linear theory of damped oscillations of continua"  ''Integral Equations, Operator Theory'' , '''1'''  (1978)  pp. 364–399; 539–566  ''Proc. Internat. Symp. Appl. Theory of Functions in Continuum Mechanics, Tbilizi'' , '''2'''  (1963)  pp. 283–322</TD></TR></table>

Revision as of 22:15, 5 June 2020


Let $ {\mathcal K} $ be a complex linear space on a which a Hermitian sesquilinear form $ [ \cdot , \cdot ] $ is defined (i.e. a mapping $ [ \cdot , \cdot ] : {\mathcal K} \times {\mathcal K} \rightarrow \mathbf C $ such that $ [ \alpha _ {1} x _ {1} + \alpha _ {2} x _ {2} , y ] = \alpha _ {1} [ x _ {1} , y ] + \alpha _ {2} [ x _ {2} , y ] $ and $ [ x , y ] = \overline{ {[ y , x ] }}\; $ for all $ x _ {1} , x _ {2} , x , y \in {\mathcal K} $, $ \alpha _ {1} , \alpha _ {2} \in \mathbf C $). Then $ {\mathcal K} $( or, more exactly, $ ( {\mathcal K} , [ \cdot , \cdot ] ) $) is called a Krein space if in $ {\mathcal K} $ there are two linear manifolds $ {\mathcal K} _ \pm $ such that

$$ \tag{a1 } {\mathcal K} = {\mathcal K} _ {+} \dot{+} {\mathcal K} _ {-} , $$

$ ( {\mathcal K} _ {+} , [ \cdot , \cdot ] ) $ and $ ( {\mathcal K} _ {-} , - [ \cdot , \cdot ] ) $ are Hilbert spaces (cf. Hilbert space) and $ [ {\mathcal K} _ {+} , {\mathcal K} _ {-} ] = \{ 0 \} $. It is always assumed that $ {\mathcal K} _ {+} , {\mathcal K} _ {-} \neq \{ 0 \} $( otherwise $ ( {\mathcal K} , [ \cdot , \cdot ] ) $ or $ ( {\mathcal K} , - [ \cdot , \cdot ] ) $ is a Hilbert space); $ [ \cdot , \cdot ] $ is called the indefinite inner product of the Krein space $ {\mathcal K} $. If, in particular, $ \kappa = \min ( \mathop{\rm dim} {\mathcal K} _ {+} , \mathop{\rm dim} {\mathcal K} _ {-} ) < \infty $, then $ {\mathcal K} _ {-} $ is a $ \pi _ \kappa $- space or Pontryagin space of index $ \kappa $( cf. also Pontryagin space); in the sequel, for a $ \pi _ \kappa $- space it is always assumed that $ \kappa = \mathop{\rm dim} {\mathcal K} _ {+} $.

Using the decomposition (a1), on the Krein space $ ( {\mathcal K} , [ \cdot , \cdot ] ) $ a Hilbert inner product $ ( \cdot , \cdot ) $ can be defined as follows:

$$ \tag{a2 } ( x , y ) = [ x _ {+} , y _ {+} ] - [ x _ {-} , y _ {-} ] , $$

$$ x = x _ {+} + x _ {-} ,\ y = y _ {+} + y _ {-} ,\ x _ \pm , y _ \pm \in {\mathcal K} _ {+} . $$

Although the decomposition (a1) is not unique, the decompositions of the components $ {\mathcal K} _ \pm $ are uniquely determined and the Hilbert norms, generated by different decompositions (a1) according to (a2), are equivalent. All topological notions in a Krein space, if not stated explicitly otherwise, refer to this topology. In the Hilbert space $ ( {\mathcal K} , ( \cdot , \cdot ) ) $, the orthogonal projections onto $ {\mathcal K} _ {+} $ and $ {\mathcal K} _ {-} $ are denoted by $ P _ {+} $ and $ P _ {-} $, respectively. Then for the operator $ J = P _ {+} - P _ {-} $, called a fundamental symmetry, one has

$$ \tag{a3 } [ x , y ] = ( J x , y ) ,\ \ x , y \in {\mathcal K} , $$

and $ J $ has the properties: $ J ^ {2} = I $, $ J = J ^ {*} $. Conversely, given a Hilbert space $ ( {\mathcal K} , ( \cdot , \cdot ) ) $ and in it an operator $ J $ with these properties (or, more generally, an operator $ G $ with $ G = G ^ {*} $, $ 0 \in \rho ( G) $), then an indefinite inner product is defined on $ {\mathcal K} $ by (a3) (or, respectively, by the relation

$$ \tag{a4 } [ x , y ] = ( G x , y ) ,\ \ x , y \in {\mathcal K} \textrm{ ) } , $$

and $ ( {\mathcal K} , [ \cdot , \cdot ] ) $ is a Krein space. Because of this construction, Krein spaces are sometimes called $ J $- spaces.

If, more generally, a Hilbert space $ ( {\mathcal H} , ( \cdot , \cdot ) ) $ and a bounded self-adjoint, not semi-definite, operator $ G $ in $ {\mathcal H} $ are given, the relation (a4) with $ x , y \in {\mathcal H} $ defines a Hermitian sesquilinear form $ [ \cdot , \cdot ] $ on $ {\mathcal H} $. This form can be extended by continuity to the completion of the quotient space $ {\mathcal H} / \mathop{\rm Ker} G $ with respect to the norm $ \| | G | ^ {1/2} x \| $( $ x \in {\mathcal H} $). This completion, equipped with $ [ \cdot , \cdot ] $, is a Krein space containing $ {\mathcal H} / \mathop{\rm Ker} G $ as a dense subset.

If $ r $ is a real and locally summable function on $ \mathbf R $ which assumes positive and negative values on sets of positive Lebesgue measure, then the space $ L _ {2,r} $ of all (classes of) measurable functions (cf. Measurable function) $ f $ on $ \mathbf R $ such that $ \int _ {- \infty } ^ \infty | f | ^ {2} | r | d x < \infty $ and equipped with the indefinite inner product $ [ f , g ] = \int _ {- \infty } ^ \infty f \overline{g}\; r d x $( $ f , g \in L _ {2,r} $) is a Krein space. More generally, if $ \sigma $ is a real function which is locally of bounded variation and not isotone on $ \mathbf R $ and $ | \sigma | $ denotes its total variation, then the space $ L _ {2} ( \sigma ) $, of all measurable functions $ f $ such that $ \int _ {- \infty } ^ \infty | f | ^ {2} d | \sigma | < \infty $ and equipped with the indefinite inner product $ [ f , g ] = \int _ {- \infty } ^ {- \infty } f \overline{g}\; d \sigma $( $ f , g \in L _ {2} ( \sigma ) $) is a Krein space.

Further, a complex linear space with a Hermitian sesquilinear form $ [ \cdot , \cdot ] $, which has $ \kappa $ negative squares (that is, each linear manifold $ {\mathcal L} \subset {\mathcal K} $ with $ [ x , x ] < 0 $ for $ x \in {\mathcal L} $, $ x \neq 0 $, is of dimension $ \leq \kappa $ and at least one such manifold is of dimension $ \kappa $), can be canonically imbedded into a $ \pi _ \kappa $- space by taking a quotient space and completing it (see [a4], [a2], [a9], [a11]).

The indefinite inner product $ [ \cdot , \cdot ] $ on the Krein space $ {\mathcal K} $ gives rise to a classification of the elements of $ {\mathcal K} $: $ x \in {\mathcal K} $ is called positive, non-negative, neutral, etc. if $ [ x , x ] > 0 $, $ [ x , x ] < 0 $, $ [ x , x ] = 0 $, etc. A linear manifold or a subspace $ {\mathcal L} $ in $ {\mathcal K} $ is called positive, non-negative, neutral, etc. if all its non-zero elements are positive, non-negative, neutral, etc. The set of all, e.g., non-negative elements is not linear, but it contains subspaces, and among them maximal ones, called maximal non-negative subspaces. All maximal non-negative subspaces of the Krein space $ {\mathcal K} $ have the same dimension (as $ {\mathcal K} _ {+} $). A subspace $ {\mathcal L} $ of $ {\mathcal K} $( with the decomposition (a1)) is maximal non-negative if and only if it can be written as $ {\mathcal L} = \{ {x _ {+} + K _ {\mathcal L} x _ {+} } : {x _ {+} \in {\mathcal K} _ {+} } \} $, where $ K _ {\mathcal L} $, the angular operator of $ {\mathcal L} $, is a contraction from $ ( {\mathcal K} _ {+} , [ \cdot , \cdot ] ) $ into $ ( {\mathcal K} _ {-} , [ \cdot , \cdot ] ) $. A dual pair $ ( {\mathcal L} _ {+} , {\mathcal L} _ {-} ) $ of subspaces of $ {\mathcal K} $ is defined as follows: $ {\mathcal L} _ {+} $ is a non-negative subspace, $ {\mathcal L} _ {-} $ is a non-positive subspace and $ [ {\mathcal L} _ {+} , {\mathcal L} _ {-} ] = \{ 0 \} $. Any dual pairs is contained in a maximal dual pair (maximality of dual pairs is defined in a natural way by inclusion); in a maximal dual pair $ ( {\mathcal L} _ {+} , {\mathcal L} _ {-} ) $ the subspace $ {\mathcal L} _ {+} $( respectively, $ {\mathcal L} _ {-} $) is maximal non-negative (respectively, non-positive) (R.S. Phillips).

Using the indefinite inner product, orthogonality can be defined in $ {\mathcal K} $: $ x , y \in {\mathcal K} $ are called orthogonal if $ [ x , y ] = 0 $; if $ {\mathcal L} \subset {\mathcal K} $, then $ {\mathcal L} ^ \perp = \{ {x } : {[ x , {\mathcal L} ] = \{ 0 \} } \} $. Some properties of orthogonality in a Hilbert space are preserved; however, there are also essential differences; e.g., $ {\mathcal L} \cap {\mathcal L} ^ \perp $ can contain non-zero vectors; $ {\mathcal L} \cap {\mathcal L} ^ \perp $ coincides with $ {\mathcal L} $ if $ {\mathcal L} $ is neutral, and $ {\mathcal L} \cap {\mathcal L} ^ \perp = \{ 0 \} $ is equivalent to $ \overline{ {{\mathcal L} + {\mathcal L} ^ \perp }}\; = {\mathcal K} $.

For a densely-defined linear operator $ T $ in the Krein space $ ( {\mathcal K} , [ \cdot , \cdot ] ) $ an adjoint $ T ^ { + } $( sometimes called $ J $- adjoint) is defined by $ [ T x , y ] = [ x , T ^ { + } y ] $( $ x \in {\mathcal D} ( T) $, $ y \in {\mathcal D} ( T ^ { + } ) $). If $ T ^ { * } $ denotes the adjoint of $ T $ in the Hilbert space $ ( {\mathcal K} , ( \cdot , \cdot ) ) $( see (a2)), then evidently $ T ^ { + } = J T ^ { * } J $. Now in the Krein space $ {\mathcal K} $ classes of operators are defined more or less similarly to the case of a Hilbert space: $ T $ is symmetric if $ T \subset T ^ { + } $, self-adjoint if $ T = T ^ { + } $, dissipative if $ \mathop{\rm Im} [ T x , x ] \geq 0 $( $ x \in {\mathcal D} ( T) $), contractive if $ [ T x , T x ] \leq [ x , x ] $( $ x \in {\mathcal K} $), unitary if $ T $ is bounded, $ {\mathcal D} ( T) = {\mathcal K} $ and $ T ^ { + } T = I = T T ^ { + } $, etc. Also, new classes of operators arise: E.g., $ T $ is a plus-operator if $ [ x , x ] \geq 0 $ implies $ [ T x , T x ] \geq 0 $, and a doubly plus-operator if $ T $ and $ T ^ { + } $ are plus-operators. In a Krein space a densely-defined isometric operator $ T $( i.e. $ [ T x , T y ] = [ x , y ] $ for all $ x , y \in {\mathcal D} ( T) $) need not be continuous. As in a Hilbert space, self-adjoint and unitary, symmetric and isometric, dissipative and contractive operators are related by the Cayley transform. E.g., if $ A = A ^ {+} $, $ z _ {0} \neq \overline{z}\; _ {0} $ and $ z _ {0} \in \rho ( A) $, then $ U = ( A - \overline{z}\; _ {0} ) ( A - z _ {0} ) ^ {-} 1 $ is unitary.

The spectrum of a self-adjoint operator $ A $ in a Krein space is not necessarily real (it can even cover the whole plane), but it is symmetric with respect to the real axis. Similarly, the spectrum of a unitary operator is symmetric with respect to the unit circle.

The indefinite inner product sometimes gives a classification of the points of the spectrum of an operator: An eigen value is said to be of positive type (negative type, etc.) if the corresponding eigen space is positive (negative, etc.).

If $ \lambda $, $ \overline \lambda \; $ are isolated eigen values of the self-adjoint operator $ A $ in a Krein space, then for the corresponding Riesz projections $ E _ \lambda $, $ E _ {\overline \lambda \; } $ one has $ E _ {\overline \lambda \; } = E _ \lambda ^ {+} $, and if, e.g., $ \mathop{\rm dim} {\mathcal R} ( E _ \lambda ) < \infty $, then the restrictions $ A \mid _ { {\mathcal R} ( E _ \lambda ) } $ and $ A \mid _ { {\mathcal R} ( E _ {\overline \lambda \; } ) } $ have the same Jordan structure. If in a $ \pi _ \kappa $- space the symmetric operator $ A $ has a real non-semi-simple eigen value $ \lambda $, then the corresponding algebraic eigen space $ {\mathcal E} _ \lambda $ can be decomposed into a direct orthogonal sum: $ {\mathcal E} _ \lambda = {\mathcal E} _ \lambda ^ { \prime } + {\mathcal E} _ \lambda ^ { \prime\prime } $, where $ {\mathcal E} _ \lambda ^ { \prime\prime } $ is a positive subspace contained in the geometric eigen space of $ A $ at $ \lambda $, and $ {\mathcal E} _ \lambda ^ { \prime } \neq \{ 0 \} $ is invariant under $ A $ with $ \mathop{\rm dim} {\mathcal E} _ \lambda ^ { \prime } < \infty $; if $ d _ {1} \dots d _ {r} $ are the lengths of the Jordan chains of $ A \mid _ { {\mathcal E} _ \lambda ^ { \prime } } $, one puts $ \rho ( \lambda )= \sum _ {j=} 1 ^ \kappa [ d _ {j} /2 ] $; if $ \lambda $ is a non-real eigen value of $ A $, one defines $ \rho ( \lambda ) $ as the dimension of the corresponding algebraic eigen space. Then $ \sum \rho ( \lambda ) \leq \kappa $, where the sum extends over all eigen values $ \lambda $ of $ A $ in the closed upper half-plane. In particular, the length of any Jordan chain of $ A $ is $ \leq 2 \kappa + 1 $, and the number of eigen values of $ A $ in the open upper half-plane, and also the number of non-semi-simple eigen values of $ A $, does not exceed $ \kappa $.

Specific results for Krein spaces are statements about the existence of maximal non-negative (or maximal non-positive) subspaces, which are invariant under a given operator. The first general result of this type was proved by L.S. Pontryagin in 1944, stating that a self-adjoint operator in a $ \pi _ \kappa $- space has a $ \kappa $- dimensional non-positive (that is, a maximal non-positive) invariant subspace. Subsequently, similar results were proved for various classes of operators in Krein spaces. E.g., a bounded linear operator $ T $ in a Krein space has a maximal non-negative invariant subspace if $ P _ {+} T P _ {-} $ is compact and, additionally, $ T $ is self-adjoint or dissipative or unitary or a plus-operator, etc. (see [a2], [a4]). One possibility for proving these results, e.g. for a unitary operator $ T $, is to establish the existence of a fixed point $ K _ {0} $ of the fractional-linear transformation

$$ K \rightarrow ( T _ {21} + T _ {22} K ) ( T _ {11} + T _ {12} K ) ^ {-} 1 , $$

where $ K $ is a contraction from $ {\mathcal K} _ {+} $ into $ {\mathcal K} _ {-} $( an angular operator) and $ ( T _ {ij} ) _ {1} ^ {2} $ is the matrix representation of $ T $ with respect to (a1). By different methods also in other cases the existence of a maximal non-negative invariant subspace has been proved, e.g.: 1) $ T $ is unitary and $ \| T ^ { n } \| $ is uniformly bounded for all $ n = 0 , 1 ,\dots $; 2) $ [ T x , T x ] > [ x , x ] $ for all $ x \in {\mathcal K} $, $ x \neq 0 $, and $ \sigma ( T) \cap \{ | \rho | = 1 \} = \emptyset $; and 3) $ T $ is bounded, self-adjoint and there exists a polynomial $ p $ such that $ [ p ( T) x , x ] \geq 0 $( $ x \in {\mathcal K} $). In many cases these maximal non-positive invariant subspaces $ {\mathcal L} $ can be specified by properties of the spectrum of $ A \mid _ {\mathcal L} $. E.g., if $ T $ is bounded, self-adjoint and $ P _ {+} T P _ {-} $ is compact, then $ {\mathcal L} $ can be chosen such that $ \mathop{\rm Im} \sigma ( A \mid _ {\mathcal L} ) \geq 0 $. There are also results about the existence of a common invariant maximal non-positive subspace for a commuting family of operators, e.g.: A commuting family of bounded self-adjoint operators in a $ \pi _ \kappa $- space has a common maximal non-negative invariant subspace (M.A. Naimark; for applications in the representation theory of groups in $ \pi _ \kappa $- spaces see [a19]). Phillips asked ([a16]) if a dual pair of subspaces of $ {\mathcal K} $ which are invariant under a commutative algebra $ A $ of bounded self-adjoint operators in the Krein space $ {\mathcal K} $ can always be extended to a maximal dual pair whose subspaces are still invariant under $ A $( which would imply that each bounded self-adjoint operator in $ {\mathcal K} $ has a maximal non-negative invariant subspace). Only partial solutions to this problem are known (cf. [a4], [a2], [a14]).

A self-adjoint operator $ A $ in the Krein space $ {\mathcal K} $ is called definitizable (positizable in [a4]) if $ \rho ( A) \neq \emptyset $ and if there exists a polynomial $ p $ such that $ [ p ( A) x , x ] \geq 0 $( $ x \in {\mathcal D} ( p ( A) ) $). Each self-adjoint operator $ A $ in a $ \pi _ \kappa $- space has this property (where $ p $ can be chosen to be $ q \overline{q}\; $ with $ q $ the minimal polynomial of $ A \mid _ {\mathcal L} $, $ {\mathcal L} $ being a $ \kappa $- dimensional non-positive invariant subspace of $ A $); also, each self-adjoint operator $ A $ in a Krein space for which $ \rho ( A ) \neq \emptyset $ and for which the Hermitian sesquilinear form $ [ A x , y ] $( $ x , y \in {\mathcal D} ( A) $) has a finite number of negative squares, is definitizable.

The non-real spectrum $ \sigma _ {0} ( A) $ of the definitizable operator $ A $ consists of at most finitely many eigen values, and $ A $ has a spectral function, with possibly certain critical points [a13], [a2]. This means that there is a finite set $ c ( A) \subset \mathbf R \cup \{ \infty \} $( of critical points) such that on the semi-ring $ \mathbf R _ {A} $, consisting of all bounded intervals of $ \mathbf R $ with end points not in $ c ( A) $ and their complements, a homomorphism $ E $ with values in the set of all self-adjoint projections in the Krein space $ {\mathcal K} $ is defined, such that for $ \Delta \in \mathbf R _ {A} $: a) $ E ( \Delta ) {\mathcal K} $ is a positive (negative) subspace if $ p > 0 $( respectively, $ p < 0 $) on $ \overline \Delta \; \cap \sigma ( A) $ for some definitizing polynomial $ p $ of $ A $; b) $ E ( \Delta ) $ is in the double commutant of the resolvent of $ A $; and c) if $ \Delta $ is bounded, then $ E ( \Delta ) {\mathcal K} \subset {\mathcal D} ( A) $ and $ \sigma ( A \mid _ {E ( \Delta ) {\mathcal K} } ) \subset \overline \Delta \; $, $ \sigma ( A \mid _ {( I - E ( \Delta ) ) {\mathcal K} } ) \subset \overline{ {( \mathbf R \setminus \Delta ) }}\; \cup \sigma _ {0} ( A) $. If, in particular, $ A $ is bounded and $ [ A x , x ] \geq 0 $( $ x \in {\mathcal K} $), then $ c ( A) \subset \{ 0 \} $, and one has

$$ A x = \int\limits _ {- \| A \| } ^ { {\| } A \| } \lambda E ( d \lambda ) x + N x , $$

for some bounded operator $ N $ such that $ N = N ^ {+} $, $ N ^ {2} = 0 $, $ [ N x , x ] \geq 0 $( $ x \in {\mathcal K} $).

If the spectrum of a definitizable operator $ A $ is discrete, then the linear span of its algebraic eigen spaces is dense in $ {\mathcal K} $; if $ A $ is compact and self-adjoint in a $ \pi _ \kappa $- space $ {\mathcal K} $ and $ 0 \notin \sigma _ {p} ( A) $, then there is a Riesz basis of $ {\mathcal K} $ consisting of eigen and associated vectors of $ A $( I.S. Iokhvidov).

There is a theory of extensions of symmetric operators to self-adjoint operators and of generalized resolvents in $ \pi _ \kappa $- spaces, and also in Krein spaces, which is similar to the Hilbert space situation. The same is true for dilation theory: Each bounded linear operator $ T $ in a Krein space $ {\mathcal K} $ has a unitary dilation $ T $ in some Krein space $ {\mathcal K} tilde \supset {\mathcal K} $([a2]). In this context one has the following result: Let $ {\mathcal K} _ {1} $, $ {\mathcal K} _ {2} $ be Krein spaces, $ {\mathcal D} $ a simply-connected open domain with smooth boundary such that $ 0 \in {\mathcal D} $, $ \overline{ {\mathcal D} }\; \subset \{ {z } : {| z | < 1 } \} $, and let $ \Theta $ be a function which is holomorphic in $ {\mathcal D} $ whose values are bounded linear operators from $ {\mathcal K} _ {1} $ to $ {\mathcal K} _ {2} $. Then there exists a Krein space $ {\mathcal K} $ and a unitary operator

$$ U = \left ( \begin{array}{ll} U _ {11} &U _ {12} \\ U _ {21} &U _ {22} \\ \end{array} \right ) : {\mathcal K} \oplus {\mathcal K} _ {1} \rightarrow {\mathcal K} \oplus {\mathcal K} _ {2} , $$

such that

$$ \Theta ( z) = U _ {22} + z U _ {21} ( I - z U _ {11} ) ^ {-} 1 U _ {12} \ \ ( z \in {\mathcal D} ) $$

(T.Ya. Azizov, see [a2], [a6]; here unitary means that $ U $ maps the Krein space $ {\mathcal K} \oplus {\mathcal K} _ {1} $ continuously onto the Krein space $ {\mathcal K} \oplus {\mathcal K} _ {2} $, preserving the indefinite inner product).

Some of the first papers about Krein spaces or, more generally, spaces with indefinite inner product, were stimulated by problems of (quantum) mechanics ([a4], [a2]; see also [a18], [a17]). Operators in Krein spaces arise also in a natural way in problems in mathematical analysis. Some examples of these are: I) Consider the canonical system of differential equations $ J \dot{x} ( t) = i H ( t) x ( t) $ on $ [ 0 , \infty ) $, where $ H ( t) $, $ J $ are $ ( n \times n ) $ matrices, $ H ( t ) \geq 0 $, $ J = J ^ {*} = J ^ {-} 1 $, and let $ U ( t) $ be the corresponding matrizant (cf. Cauchy operator): $ J \dot{U} ( t) = i H ( t) U ( t) $, $ U ( 0) = I _ {n} $. Then $ U ( t) $ is $ J $- unitary (that is, unitary with respect to the inner product defined in $ \mathbf C ^ {n} $ by the matrix $ J $, see (a3)), and, e.g., in the stability theory for periodic equations $ ( H ( t) = H ( T + t ) ) $ the classification of the eigen values of $ U ( T) $ into those of positive or negative type plays an essential role ([a5], [a8]). II) The integral operator $ x ( \cdot ) \rightarrow \int _ {a} ^ {b} K ( \cdot , s ) x ( s ) d \sigma ( s) $, $ \sigma $ real and of bounded variation on the interval $ [ a , b ] $, $ K ( s , t ) = \overline{ {K ( t , s ) }}\; $( $ s , t \in [ a , b ] $), is self-adjoint in the Krein space $ L _ {2} ( \sigma ) $. III) The theory of dual pairs of subspaces of a Krein space and their extensions to maximal dual pairs is related to certain questions in the theory of extensions of dissipative operators in a Hilbert space to maximal dissipative ones. Phillips started these investigations in connection with the Cauchy problem for dissipative hyperbolic and parabolic systems (see [a2], [a4] for references). IV) With the monic operator polynomial $ L ( \lambda ) = \lambda ^ {n} I + \lambda ^ {n-} 1 B _ {n-} 1 + \dots + \lambda B _ {1} + B _ {0} $, $ B _ {j} $ bounded self-adjoint operators in some Hilbert space $ {\mathcal H} $, one can associate the so-called companion operator

$$ A = \left ( \begin{array}{ccccc} - B _ {n-} 1 &- B _ {n-} 2 &\dots &- B _ {1} &- B _ {0} \\ I & 0 &\dots & 0 & 0 \\ \cdot &\cdot &\dots &\cdot &\cdot \\ 0 & 0 &\dots & I & 0 \\ \end{array} \right ) , $$

which is self-adjoint in the Krein space $ {\mathcal K} = {\mathcal H} ^ {n} $, $ [ x , y ] = ( G x , y ) $( $ x , y \in {\mathcal H} ^ {n} $), where $ ( \cdot , \cdot ) $ is the inner product in $ {\mathcal H} ^ {n} $ and

$$ G = \left ( \begin{array}{cccc} 0 & 0 &\dots & I \\ 0 & 0 &{} &B _ {n-} 1 \\ \cdot &\cdot &{} &\cdot \\ \cdot &\cdot &{} &\cdot \\ \cdot &\cdot &{} &\cdot \\ I &B _ {n-} 1 &\dots &B _ {1} \\ \end{array} \right ) . $$

If, e.g., $ n = 2 $ and $ B _ {0} $ is compact and $ \geq 0 $, the results about the existence of maximal non-negative invariant subspaces mentioned above imply that there exists a bounded linear operator $ Z $ in $ {\mathcal H} $ satisfying $ Z ^ {2} + B _ {1} Z + B _ {0} = 0 $, $ Z ^ {*} Z \leq B _ {0} $ and $ \mathop{\rm Im} \sigma ( Z) \geq 0 $[a12]. In a similar way, if $ B \geq 0 $, $ C = C ^ {*} $ and $ A $ are $ ( n \times n ) $ matrices such that $ G = ( _ { A } ^ {-} C {} _ {B} ^ {A ^ {*} } ) $ has $ n $ positive and $ n $ negative eigen values, the solutions $ X $ of the matrix Riccati equation

$$ X B X + X A + A ^ {*} X - C = 0 $$

with $ ( X ^ {*} - X ) ( A + B X ) \geq 0 $ are in bijective correspondence with all maximal non-positive subspaces which are invariant under the self-adjoint operator $ T = i ( _ {C} ^ {A} {} _ {- A ^ {*} } ^ { B } ) $ in the $ 2n $- dimensional Krein space $ {\mathcal K} = \mathbf C ^ {2n} $, equipped with the indefinite inner product (a4) (see [a8]). V) If $ L $ is a formally-symmetric regular ordinary differential operator on the interval $ [ a , b ] $ with symmetric boundary conditions at $ a $ and $ b $, and $ r $ is a summable function on $ [ a , b ] $ which is not of constant sign (a.e.) on $ [ a , b ] $, then the differential equation $ L y - \lambda r y = r f $ leads to a self-adjoint operator $ A $ in the Krein space $ {\mathcal K} = L _ {2,r} $ with inner product $ [ f , g ] = \int _ {a} ^ {b} f \overline{g}\; r d x $. If $ L $ is semi-bounded from below, the operator $ A $ is definitizable. VI) Krein spaces can be associated with certain eigen value problems for ordinary differential operators containing the eigen value parameters in the boundary conditions. E.g., consider in $ L _ {2} = L _ {2} [ 0 , \infty ) $ the problem

$$ - \frac{d ^ {2} y }{d x ^ {2} } + q y - \lambda y = f , $$

which is supposed to have a limit point at $ \infty $ and with a boundary condition $ \alpha ( \lambda ) y ( 0) + \beta ( \lambda ) y ^ \prime ( 0) = 0 $ at $ x = 0 $( $ \alpha $, $ \beta $ are functions which are holomorphic on some set $ D _ {\alpha , \beta } \subset \mathbf C $ and satisfying a symmetry condition). The solution of this problem can be represented as $ y = P ( A - \lambda I ) ^ {-} 1 f $( $ f \in L _ {2} $), where, in general, $ A $ is a self-adjoint operator in some Krein space $ {\mathcal K} = L _ {2} \oplus {\mathcal K} _ {1} $ and $ P $ is the orthogonal projection from $ {\mathcal K} $ onto $ L _ {2} $[a17]. VII) Certain classes of analytic functions are closely related to the theory of operators in $ \pi _ \kappa $- spaces. This concerns, e.g., functions $ f $ which are defined and meromorphic in the upper half-plane (or the unit disc) and which are such that the kernel

$$ N _ {f} ( z , \rho ) = \ \frac{f ( z) - \overline{ {f ( \rho ) }}\; }{z - \overline \rho \; } $$

(or

$$ \left . S _ {f} ( z , \overline \rho \; ) = \ \frac{1 - f ( z) \overline{ {f ( \rho ) }}\; }{1 - z \overline \rho \; } \right ) $$

has $ \kappa $ negative squares (that is, for arbitrary $ n $ and $ z _ {1} \dots z _ {n} $, the matrix $ ( N _ {f} ( z _ {i} , z _ {j} ) ) _ {1} ^ {n} $ has at most $ n $ negative eigen values and for at least one choice of $ n , z _ {1} \dots z _ {n} $ it has $ \kappa $ negative eigen values). Corresponding extrapolation or moment problems can be treated by making use of results of the theory of symmetric or isometric operators in $ \pi _ \kappa $- spaces (see [a12], [a2]).

References

[a1] T.Ya Azizov, I.S. Iokhvidov, "Linear operators in spaces with indefinite metric and their applications" Russian Math. Surveys , 15 (1981) pp. 438–490 Itogi Nauk. i Tekhn. Mat. Anal. , 17 (1979) pp. 113–205
[a2] T.Ya Azizov, I.S. Iokhvidov, "Foundations of the theory of linear operators in spaces with indefinite metric" , Moscow (1986) (In Russian)
[a3] T. Ando, "Linear operators in Krein spaces" , Hokkaido Univ. (1979)
[a4] J. Bognár, "Indefinite inner product spaces" , Springer (1974)
[a5] Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)
[a6] A. Dijksma, H. Langer, H.S.V. de Snoo, "Unitary colligations in Krein spaces and their role in the extension theory of isometries and symmetric linear relations in Hilbert spaces" S. Kurepa (ed.) et al. (ed.) , Foundational analysis II , Lect. notes in math. , 1247 , Springer (1987) pp. 1–42
[a7] A. Dijksma, H. Langer, H.S.V. de Snoo, "Symmetric Sturm–Liouville operators with eigenvalues depending boundary conditions" , Oscillation, Bifurcations and Chaos , CMS Conf. Proc. , 8 , Amer. Math. Soc. (1987) pp. 87–116
[a8] I. [I. Gokhberg] Gohberg, P. Lancaster, L. Rodman, "Matrices and indefinite scalar products" , Birkhäuser (1983)
[a9] I.S. [I.S. Iokhvidov] Iohidov, M.G. Krein, H. Langer, "Introduction to the spectral theory of operators in spaces with an indefinite metric" , Akademie Verlag (1982)
[a10] V.I. Istraţescu, "Inner product spaces. Theory and applications" , Reidel (1987)
[a11] M.G. Krein, "Introduction to the geometry of indefinite -spaces and the theory of operators in these spaces" , Second Math. Summer School , 1 , Kiev (1965) pp. 15–92 (In Russian)
[a12] M.G. Krein, H. Langer, "Ueber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume zusammenhängen, I: Einige Funktionenklassen und ihre Darstellungen" Math. Nachr. , 77 (1977) pp. 187–236
[a13] H. Langer, "Spectral functions of definitizable operators in Krein spaces" D. Butković (ed.) et al. (ed.) , Functional analysis , Lect. notes in math. , 948 , Springer (1982) pp. 1–46
[a14] H. Langer, "Invariante Teilräume definisierbarer -selbstadjungierter Operatoren" Ann. Acad. Sci. Fenn A. I , 475 (1971)
[a15] J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973)
[a16] R.S. Phillips, "The extensions of dual subspaces invariant under an algebra" , Proc. Internat. Symp. Linear Spaces (Jerusalem, 1960) , Pergamon (1961) pp. 366–398
[a17] L. Bracci, G. Morchio, F. Strocchi, "Wigner's theorem on symmetries in indefinite metric spaces" Comm. Math. Phys. , 41 (1975) pp. 289–299
[a18] K.L. Nagy, "State vector spaces with indefinite metric in quantum field theory" , Noordhoff (1966)
[a19] M.A. Naimark, R.S. Ismagilov, "Representations of groups and algebras in a space with indefinite metric" Itogi Nauk. i Tekhn. Mat. Anal. (1969) pp. 73–105 (In Russian)
[a20] M.G. Krein, H. Langer, "On some mathematical principles in the linear theory of damped oscillations of continua" Integral Equations, Operator Theory , 1 (1978) pp. 364–399; 539–566 Proc. Internat. Symp. Appl. Theory of Functions in Continuum Mechanics, Tbilizi , 2 (1963) pp. 283–322
How to Cite This Entry:
Krein space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Krein_space&oldid=47525
This article was adapted from an original article by H. Langer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article