# Krein space

Let ${\mathcal K}$ be a complex linear space on 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 $G = \left({\begin{array}{cc} -C & A^* \\ A & B\end{array}}\right)$ 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 $T = i \left({\begin{array}{cc} A & B \\ C & -A^*\end{array}}\right)$ 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 $J$-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 $\pi _ { \kappa}$ 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 $J$-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=55242
This article was adapted from an original article by H. Langer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article