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A [[Hilbert space with an indefinite metric|Hilbert space with an indefinite metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738001.png" /> that has a finite rank of indefiniteness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738002.png" />. Basic facts concerning the geometry of these spaces were established by L.S. Pontryagin [[#References|[1]]]. Besides the facts common for spaces with an indefinite metric, the following properties hold.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738003.png" /> is an arbitrary non-negative linear manifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738004.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738005.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738006.png" /> is a positive linear manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738007.png" />, then its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738008.png" />-orthogonal complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738009.png" /> is a negative linear manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380010.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380011.png" /> is a complete space with respect to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380012.png" />. If the linear manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380013.png" /> is non-degenerate, then its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380014.png" />-orthogonal complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380015.png" /> is non-degenerate as well and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380016.png" />.
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The spectrum (in particular, the discrete spectrum) of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380017.png" />-unitary (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380018.png" />-self-adjoint) operator is symmetric with respect to the unit circle (real line), all elementary divisors corresponding to eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380020.png" />, are of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380023.png" />. The sum of the dimensions of the root subspaces of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380024.png" />-unitary (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380025.png" />-self-adjoint) operator corresponding to eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380027.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380028.png" />), does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380029.png" />.
+
A [[Hilbert space with an indefinite metric|Hilbert space with an indefinite metric]]  $  \Pi _  \kappa  $
 +
that has a finite rank of indefiniteness  $  \kappa $.  
 +
Basic facts concerning the geometry of these spaces were established by L.S. Pontryagin [[#References|[1]]]. Besides the facts common for spaces with an indefinite metric, the following properties hold.
  
The following theorem [[#References|[1]]] is fundamental in the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380031.png" />-self-adjoint operators on a Pontryagin space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380032.png" />: For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380033.png" />-self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380035.png" />) there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380036.png" />-dimensional (maximal) non-negative invariant subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380037.png" /> in which all eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380038.png" /> have non-negative imaginary parts, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380039.png" />-dimensional non-negative invariant subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380040.png" /> in which all eigen values have non-positive imaginary parts. A similar statement in which the upper (lower) half-plane is replaced by the exterior (interior) of the unit disc is also valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380041.png" />-unitary operators, and under certain additional conditions — even for operators on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380042.png" />.
+
If  $  {\mathcal P} $
 +
is an arbitrary non-negative linear manifold in $  \Pi _  \kappa  $,  
 +
then  $  \mathop{\rm dim}  {\mathcal P} \leq  \kappa $;
 +
if  $  {\mathcal P} $
 +
is a positive linear manifold and $  \mathop{\rm dim}  {\mathcal P} = \kappa $,
 +
then its  $  J $-
 +
orthogonal complement  $  N $
 +
is a negative linear manifold and  $  \Pi _  \kappa  = {\mathcal P} \oplus N $.  
 +
Moreover,  $  N $
 +
is a complete space with respect to the norm  $  | x | = \sqrt {- J ( x , x ) } $.
 +
If the linear manifold  $  L \subset  \Pi _  \kappa  $
 +
is non-degenerate, then its  $  J $-
 +
orthogonal complement  $  M $
 +
is non-degenerate as well and $  \Pi _  \kappa  = M \oplus L $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380043.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380045.png" />-unitary operator, then its maximal invariant subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380047.png" /> can be chosen so that the elementary divisors of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380049.png" /> are of minimal order. In order that a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380050.png" /> with no roots inside the unit disc has the property: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380052.png" />, it is necessary and sufficient that it can be divided by the minimal annihilating polynomial of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380053.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380054.png" /> is a cyclic operator, then its non-negative invariant subspaces of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380055.png" /> are uniquely determined. In this case the above-mentioned property of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380056.png" /> with roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380057.png" /> outside the unit disc, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380058.png" />, is equivalent to the divisibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380059.png" /> by the characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380060.png" />.
+
The spectrum (in particular, the discrete spectrum) of a $  J $-
 +
unitary ( $  J $-
 +
self-adjoint) operator is symmetric with respect to the unit circle (real line), all elementary divisors corresponding to eigen values  $  \lambda $,
 +
$  | \lambda | > 1 $,  
 +
are of finite order $  \rho _  \lambda  $,
 +
$  \rho _  \lambda  \leq  \kappa $,  
 +
$  \rho _  \lambda  = - \rho _ {\lambda  ^ {-}  1 } $.  
 +
The sum of the dimensions of the root subspaces of a $  J $-
 +
unitary ( $  J $-
 +
self-adjoint) operator corresponding to eigen values  $  \lambda $,  
 +
$  | \lambda | > 1 $(
 +
$  \mathop{\rm lm}  \lambda > 0 $),  
 +
does not exceed  $  \kappa $.
  
Each completely-continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380061.png" />-self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380062.png" /> on a Pontryagin space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380063.png" /> such that zero belongs to its continuous spectrum does not have a residual spectrum. The root vectors of such an operator form a [[Riesz basis|Riesz basis]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380064.png" /> with respect to the (definite) norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380065.png" />.
+
The following theorem [[#References|[1]]] is fundamental in the theory of  $  J $-
 +
self-adjoint operators on a Pontryagin space $  \Pi _  \kappa  $:
 +
For each  $  J $-
 +
self-adjoint operator  $  A $(
 +
$  \overline{ {D ( A) }}\; = \Pi _  \kappa  $)
 +
there exists a $  \kappa $-
 +
dimensional (maximal) non-negative invariant subspace  $  {\mathcal T} $
 +
in which all eigen values of $  A $
 +
have non-negative imaginary parts, and a $  \kappa $-
 +
dimensional non-negative invariant subspace  $  {\mathcal T}  ^  \prime  $
 +
in which all eigen values have non-positive imaginary parts. A similar statement in which the upper (lower) half-plane is replaced by the exterior (interior) of the unit disc is also valid for  $  J $-
 +
unitary operators, and under certain additional conditions — even for operators on the space  $  \Pi _  \infty  $.
  
Many facts concerning invariant subspaces and the spectrum can be generalized to a case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380067.png" />-isometric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380069.png" />-non-expanding operators. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380070.png" /> is an arbitrary set of eigen values of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380071.png" />-isometric operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380073.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380074.png" /> is the order of the elementary divisor at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380076.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380077.png" />-non-expanding boundedly-invertible operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380078.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380079.png" />-dimensional invariant non-negative subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380080.png" /> such that all eigen values of the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380081.png" /> lie in the unit disc [[#References|[2]]]. A similar fact holds for maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380083.png" />-dissipative operators. In general, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380084.png" />-dissipative operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380086.png" />, has at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380087.png" /> eigen values in the upper half-plane. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380088.png" />-isometric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380089.png" />-symmetric (and more generally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380090.png" />-non-expanding and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380091.png" />-dissipative) operators are related by the Cayley transformation (cf. [[Cayley transform|Cayley transform]]), which has on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380092.png" /> all natural properties [[#References|[2]]]. This fact allows one to develop the extension theory simultaneously for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380093.png" />-isometric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380094.png" />-symmetric operators. In particular, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380095.png" />-isometric (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380096.png" />-symmetric) operator can be extended to a maximal one. If its deficiency indices are different, then it has no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380097.png" />-unitary (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380098.png" />-self-adjoint) extensions. If these indices are equal and finite, then any maximal extension is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380099.png" />-unitary (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800100.png" />-self-adjoint).
+
If  $  U $
 +
is a  $  J $-
 +
unitary operator, then its maximal invariant subspaces $  {\mathcal T} $,
 +
$  {\mathcal T}  ^  \prime  $
 +
can be chosen so that the elementary divisors of the operator  $  U _  {\mathcal T}  = U \mid  _  {\mathcal T}  $,  
 +
$  U _ { {\mathcal T}  ^  \prime  } = U \mid  _ { {\mathcal T}  ^  \prime  } $
 +
are of minimal order. In order that a polynomial  $  P ( \lambda ) $
 +
with no roots inside the unit disc has the property: $  ( P ( U) x , P ( U) x ) \leq  0 $,  
 +
$  x \in \Pi _  \kappa  $,  
 +
it is necessary and sufficient that it can be divided by the minimal annihilating polynomial of the operator  $  U _  {\mathcal T}  $.  
 +
If  $  U $
 +
is a cyclic operator, then its non-negative invariant subspaces of dimension  $  \kappa $
 +
are uniquely determined. In this case the above-mentioned property of the polynomial  $  P $
 +
with roots  $  \{ \lambda _ {i} \} $
 +
outside the unit disc,  $  | \lambda _ {i} | > 1 $,  
 +
is equivalent to the divisibility of  $  P ( \lambda ) $
 +
by the characteristic polynomial of  $  U _  {\mathcal T}  $.
  
For completely-continuous operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800101.png" />, a number of statements on the completeness of the system of root vectors, analogous to the corresponding facts from the theory of dissipative operators on spaces with a definite metric, is valid.
+
Each completely-continuous $  J $-
 +
self-adjoint operator  $  A $
 +
on a Pontryagin space  $  \Pi _  \kappa  $
 +
such that zero belongs to its continuous spectrum does not have a residual spectrum. The root vectors of such an operator form a [[Riesz basis|Riesz basis]] in  $  \Pi _  \kappa  $
 +
with respect to the (definite) norm  $  ( | J | x , x ) $.
 +
 
 +
Many facts concerning invariant subspaces and the spectrum can be generalized to a case of  $  J $-
 +
isometric and  $  J $-
 +
non-expanding operators. Thus, if  $  \lambda _ {1} \dots \lambda _ {n} $
 +
is an arbitrary set of eigen values of a  $  J $-
 +
isometric operator,  $  \lambda _ {i} \overline \lambda \; _ {k} \neq 1 $,
 +
$  i , k = 1 \dots n $,
 +
and if  $  \rho _ {i} $
 +
is the order of the elementary divisor at the point  $  \lambda _ {i} $,
 +
then  $  \sum _ {1}  ^ {n} \rho _ {i} \leq  \kappa $.
 +
Any  $  J $-
 +
non-expanding boundedly-invertible operator  $  T $
 +
has a  $  \kappa $-
 +
dimensional invariant non-negative subspace  $  {\mathcal T} $
 +
such that all eigen values of the restriction  $  T \mid  _  {\mathcal T}  $
 +
lie in the unit disc [[#References|[2]]]. A similar fact holds for maximal  $  J $-
 +
dissipative operators. In general, a  $  J $-
 +
dissipative operator  $  A $,
 +
$  D ( A) \subset  D ( A  ^ {*} ) $,
 +
has at most  $  \kappa $
 +
eigen values in the upper half-plane.  $  J $-
 +
isometric and  $  J $-
 +
symmetric (and more generally,  $  J $-
 +
non-expanding and  $  J $-
 +
dissipative) operators are related by the Cayley transformation (cf. [[Cayley transform|Cayley transform]]), which has on $  \Pi _  \kappa  $
 +
all natural properties [[#References|[2]]]. This fact allows one to develop the extension theory simultaneously for  $  J $-
 +
isometric and  $  J $-
 +
symmetric operators. In particular, every  $  J $-
 +
isometric ( $  J $-
 +
symmetric) operator can be extended to a maximal one. If its deficiency indices are different, then it has no  $  J $-
 +
unitary ( $  J $-
 +
self-adjoint) extensions. If these indices are equal and finite, then any maximal extension is  $  J $-
 +
unitary ( $  J $-
 +
self-adjoint).
 +
 
 +
For completely-continuous operators on  $  \Pi _  \kappa  $,  
 +
a number of statements on the completeness of the system of root vectors, analogous to the corresponding facts from the theory of dissipative operators on spaces with a definite metric, is valid.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Hermitian operators in a space with indefinite metric"  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''8'''  (1944)  pp. 243–280  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.S. Iokhvidov,  M.G. Krein,  "Spectral theory of operators in a space with indefinite metric I"  ''Transl. Amer. Math. Soc. (2)'' , '''13'''  (1960)  pp. 105–175  ''Trudy Moskov. Mat. Obshch.'' , '''5'''  (1956)  pp. 367–432</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.S. Iokhvidov,  M.G. Krein,  "Spectral theory of operators in a space with indefinite metric II"  ''Trudy Moskov. Mat. Obshch.'' , '''8'''  (1959)  pp. 413–496  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  T.Ya. Azizov,  I.S. Iokhvidov,  "Linear operators in Hilbert spaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800102.png" />-metric"  ''Russian Math. Surveys'' , '''26''' :  4  (1971)  pp. 45–97  ''Uspekhi Mat. Nauk'' , '''26''' :  4  (1971)  pp. 43–92</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.G. Krein,  "Introduction to the geometry of indefinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800103.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">[6]</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">[7]</TD> <TD valign="top">  L. Nagy,  "State vector spaces with indefinite metric in quantum field theory" , Noordhoff  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Hermitian operators in a space with indefinite metric"  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''8'''  (1944)  pp. 243–280  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.S. Iokhvidov,  M.G. Krein,  "Spectral theory of operators in a space with indefinite metric I"  ''Transl. Amer. Math. Soc. (2)'' , '''13'''  (1960)  pp. 105–175  ''Trudy Moskov. Mat. Obshch.'' , '''5'''  (1956)  pp. 367–432</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.S. Iokhvidov,  M.G. Krein,  "Spectral theory of operators in a space with indefinite metric II"  ''Trudy Moskov. Mat. Obshch.'' , '''8'''  (1959)  pp. 413–496  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  T.Ya. Azizov,  I.S. Iokhvidov,  "Linear operators in Hilbert spaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800102.png" />-metric"  ''Russian Math. Surveys'' , '''26''' :  4  (1971)  pp. 45–97  ''Uspekhi Mat. Nauk'' , '''26''' :  4  (1971)  pp. 43–92</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.G. Krein,  "Introduction to the geometry of indefinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800103.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">[6]</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">[7]</TD> <TD valign="top">  L. Nagy,  "State vector spaces with indefinite metric in quantum field theory" , Noordhoff  (1966)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Pontryagin spaces form a subclass of the class of Krein spaces (cf. [[Krein space|Krein space]] and also [[Hilbert space with an indefinite metric|Hilbert space with an indefinite metric]]). The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800104.png" /> appearing in the beginning of the main article above is the fundamental symmetry (see [[Krein space|Krein space]]), which defines the indefinite inner product via the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800105.png" />.
+
Pontryagin spaces form a subclass of the class of Krein spaces (cf. [[Krein space|Krein space]] and also [[Hilbert space with an indefinite metric|Hilbert space with an indefinite metric]]). The operator $  J $
 +
appearing in the beginning of the main article above is the fundamental symmetry (see [[Krein space|Krein space]]), which defines the indefinite inner product via the formula $  [ x, y] = ( Jx, y) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.Ya. Azizov,  I.S. [I.S. Iokhvidov] Iohidov,  "Linear operators in spaces with an indefinite metric" , Wiley  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</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></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.Ya. Azizov,  I.S. [I.S. Iokhvidov] Iohidov,  "Linear operators in spaces with an indefinite metric" , Wiley  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</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></table>

Latest revision as of 08:07, 6 June 2020


A Hilbert space with an indefinite metric $ \Pi _ \kappa $ that has a finite rank of indefiniteness $ \kappa $. Basic facts concerning the geometry of these spaces were established by L.S. Pontryagin [1]. Besides the facts common for spaces with an indefinite metric, the following properties hold.

If $ {\mathcal P} $ is an arbitrary non-negative linear manifold in $ \Pi _ \kappa $, then $ \mathop{\rm dim} {\mathcal P} \leq \kappa $; if $ {\mathcal P} $ is a positive linear manifold and $ \mathop{\rm dim} {\mathcal P} = \kappa $, then its $ J $- orthogonal complement $ N $ is a negative linear manifold and $ \Pi _ \kappa = {\mathcal P} \oplus N $. Moreover, $ N $ is a complete space with respect to the norm $ | x | = \sqrt {- J ( x , x ) } $. If the linear manifold $ L \subset \Pi _ \kappa $ is non-degenerate, then its $ J $- orthogonal complement $ M $ is non-degenerate as well and $ \Pi _ \kappa = M \oplus L $.

The spectrum (in particular, the discrete spectrum) of a $ J $- unitary ( $ J $- self-adjoint) operator is symmetric with respect to the unit circle (real line), all elementary divisors corresponding to eigen values $ \lambda $, $ | \lambda | > 1 $, are of finite order $ \rho _ \lambda $, $ \rho _ \lambda \leq \kappa $, $ \rho _ \lambda = - \rho _ {\lambda ^ {-} 1 } $. The sum of the dimensions of the root subspaces of a $ J $- unitary ( $ J $- self-adjoint) operator corresponding to eigen values $ \lambda $, $ | \lambda | > 1 $( $ \mathop{\rm lm} \lambda > 0 $), does not exceed $ \kappa $.

The following theorem [1] is fundamental in the theory of $ J $- self-adjoint operators on a Pontryagin space $ \Pi _ \kappa $: For each $ J $- self-adjoint operator $ A $( $ \overline{ {D ( A) }}\; = \Pi _ \kappa $) there exists a $ \kappa $- dimensional (maximal) non-negative invariant subspace $ {\mathcal T} $ in which all eigen values of $ A $ have non-negative imaginary parts, and a $ \kappa $- dimensional non-negative invariant subspace $ {\mathcal T} ^ \prime $ in which all eigen values have non-positive imaginary parts. A similar statement in which the upper (lower) half-plane is replaced by the exterior (interior) of the unit disc is also valid for $ J $- unitary operators, and under certain additional conditions — even for operators on the space $ \Pi _ \infty $.

If $ U $ is a $ J $- unitary operator, then its maximal invariant subspaces $ {\mathcal T} $, $ {\mathcal T} ^ \prime $ can be chosen so that the elementary divisors of the operator $ U _ {\mathcal T} = U \mid _ {\mathcal T} $, $ U _ { {\mathcal T} ^ \prime } = U \mid _ { {\mathcal T} ^ \prime } $ are of minimal order. In order that a polynomial $ P ( \lambda ) $ with no roots inside the unit disc has the property: $ ( P ( U) x , P ( U) x ) \leq 0 $, $ x \in \Pi _ \kappa $, it is necessary and sufficient that it can be divided by the minimal annihilating polynomial of the operator $ U _ {\mathcal T} $. If $ U $ is a cyclic operator, then its non-negative invariant subspaces of dimension $ \kappa $ are uniquely determined. In this case the above-mentioned property of the polynomial $ P $ with roots $ \{ \lambda _ {i} \} $ outside the unit disc, $ | \lambda _ {i} | > 1 $, is equivalent to the divisibility of $ P ( \lambda ) $ by the characteristic polynomial of $ U _ {\mathcal T} $.

Each completely-continuous $ J $- self-adjoint operator $ A $ on a Pontryagin space $ \Pi _ \kappa $ such that zero belongs to its continuous spectrum does not have a residual spectrum. The root vectors of such an operator form a Riesz basis in $ \Pi _ \kappa $ with respect to the (definite) norm $ ( | J | x , x ) $.

Many facts concerning invariant subspaces and the spectrum can be generalized to a case of $ J $- isometric and $ J $- non-expanding operators. Thus, if $ \lambda _ {1} \dots \lambda _ {n} $ is an arbitrary set of eigen values of a $ J $- isometric operator, $ \lambda _ {i} \overline \lambda \; _ {k} \neq 1 $, $ i , k = 1 \dots n $, and if $ \rho _ {i} $ is the order of the elementary divisor at the point $ \lambda _ {i} $, then $ \sum _ {1} ^ {n} \rho _ {i} \leq \kappa $. Any $ J $- non-expanding boundedly-invertible operator $ T $ has a $ \kappa $- dimensional invariant non-negative subspace $ {\mathcal T} $ such that all eigen values of the restriction $ T \mid _ {\mathcal T} $ lie in the unit disc [2]. A similar fact holds for maximal $ J $- dissipative operators. In general, a $ J $- dissipative operator $ A $, $ D ( A) \subset D ( A ^ {*} ) $, has at most $ \kappa $ eigen values in the upper half-plane. $ J $- isometric and $ J $- symmetric (and more generally, $ J $- non-expanding and $ J $- dissipative) operators are related by the Cayley transformation (cf. Cayley transform), which has on $ \Pi _ \kappa $ all natural properties [2]. This fact allows one to develop the extension theory simultaneously for $ J $- isometric and $ J $- symmetric operators. In particular, every $ J $- isometric ( $ J $- symmetric) operator can be extended to a maximal one. If its deficiency indices are different, then it has no $ J $- unitary ( $ J $- self-adjoint) extensions. If these indices are equal and finite, then any maximal extension is $ J $- unitary ( $ J $- self-adjoint).

For completely-continuous operators on $ \Pi _ \kappa $, a number of statements on the completeness of the system of root vectors, analogous to the corresponding facts from the theory of dissipative operators on spaces with a definite metric, is valid.

References

[1] L.S. Pontryagin, "Hermitian operators in a space with indefinite metric" Izv. Akad. Nauk. SSSR Ser. Mat. , 8 (1944) pp. 243–280 (In Russian)
[2] I.S. Iokhvidov, M.G. Krein, "Spectral theory of operators in a space with indefinite metric I" Transl. Amer. Math. Soc. (2) , 13 (1960) pp. 105–175 Trudy Moskov. Mat. Obshch. , 5 (1956) pp. 367–432
[3] I.S. Iokhvidov, M.G. Krein, "Spectral theory of operators in a space with indefinite metric II" Trudy Moskov. Mat. Obshch. , 8 (1959) pp. 413–496 (In Russian)
[4] T.Ya. Azizov, I.S. Iokhvidov, "Linear operators in Hilbert spaces with -metric" Russian Math. Surveys , 26 : 4 (1971) pp. 45–97 Uspekhi Mat. Nauk , 26 : 4 (1971) pp. 43–92
[5] 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)
[6] 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)
[7] L. Nagy, "State vector spaces with indefinite metric in quantum field theory" , Noordhoff (1966)

Comments

Pontryagin spaces form a subclass of the class of Krein spaces (cf. Krein space and also Hilbert space with an indefinite metric). The operator $ J $ appearing in the beginning of the main article above is the fundamental symmetry (see Krein space), which defines the indefinite inner product via the formula $ [ x, y] = ( Jx, y) $.

References

[a1] T.Ya. Azizov, I.S. [I.S. Iokhvidov] Iohidov, "Linear operators in spaces with an indefinite metric" , Wiley (1989) (Translated from Russian)
[a2] 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)
How to Cite This Entry:
Pontryagin space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_space&oldid=11492
This article was adapted from an original article by N.K. Nikol'skiiB.S. Pavlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article