Krein space
Let be a complex linear space on a which a Hermitian sesquilinear form
is defined (i.e. a mapping
such that
and
for all
,
). Then
(or, more exactly,
) is called a Krein space if in
there are two linear manifolds
such that
![]() | (a1) |
and
are Hilbert spaces (cf. Hilbert space) and
. It is always assumed that
(otherwise
or
is a Hilbert space);
is called the indefinite inner product of the Krein space
. If, in particular,
, then
is a
-space or Pontryagin space of index
(cf. also Pontryagin space); in the sequel, for a
-space it is always assumed that
.
Using the decomposition (a1), on the Krein space a Hilbert inner product
can be defined as follows:
![]() | (a2) |
![]() |
Although the decomposition (a1) is not unique, the decompositions of the components 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
, the orthogonal projections onto
and
are denoted by
and
, respectively. Then for the operator
, called a fundamental symmetry, one has
![]() | (a3) |
and has the properties:
,
. Conversely, given a Hilbert space
and in it an operator
with these properties (or, more generally, an operator
with
,
), then an indefinite inner product is defined on
by (a3) (or, respectively, by the relation
![]() | (a4) |
and is a Krein space. Because of this construction, Krein spaces are sometimes called
-spaces.
If, more generally, a Hilbert space and a bounded self-adjoint, not semi-definite, operator
in
are given, the relation (a4) with
defines a Hermitian sesquilinear form
on
. This form can be extended by continuity to the completion of the quotient space
with respect to the norm
(
). This completion, equipped with
, is a Krein space containing
as a dense subset.
If is a real and locally summable function on
which assumes positive and negative values on sets of positive Lebesgue measure, then the space
of all (classes of) measurable functions (cf. Measurable function)
on
such that
and equipped with the indefinite inner product
(
) is a Krein space. More generally, if
is a real function which is locally of bounded variation and not isotone on
and
denotes its total variation, then the space
, of all measurable functions
such that
and equipped with the indefinite inner product
(
) is a Krein space.
Further, a complex linear space with a Hermitian sesquilinear form , which has
negative squares (that is, each linear manifold
with
for
,
, is of dimension
and at least one such manifold is of dimension
), can be canonically imbedded into a
-space by taking a quotient space and completing it (see [a4], [a2], [a9], [a11]).
The indefinite inner product on the Krein space
gives rise to a classification of the elements of
:
is called positive, non-negative, neutral, etc. if
,
,
, etc. A linear manifold or a subspace
in
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
have the same dimension (as
). A subspace
of
(with the decomposition (a1)) is maximal non-negative if and only if it can be written as
, where
, the angular operator of
, is a contraction from
into
. A dual pair
of subspaces of
is defined as follows:
is a non-negative subspace,
is a non-positive subspace and
. 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
the subspace
(respectively,
) is maximal non-negative (respectively, non-positive) (R.S. Phillips).
Using the indefinite inner product, orthogonality can be defined in :
are called orthogonal if
; if
, then
. Some properties of orthogonality in a Hilbert space are preserved; however, there are also essential differences; e.g.,
can contain non-zero vectors;
coincides with
if
is neutral, and
is equivalent to
.
For a densely-defined linear operator in the Krein space
an adjoint
(sometimes called
-adjoint) is defined by
(
,
). If
denotes the adjoint of
in the Hilbert space
(see (a2)), then evidently
. Now in the Krein space
classes of operators are defined more or less similarly to the case of a Hilbert space:
is symmetric if
, self-adjoint if
, dissipative if
(
), contractive if
(
), unitary if
is bounded,
and
, etc. Also, new classes of operators arise: E.g.,
is a plus-operator if
implies
, and a doubly plus-operator if
and
are plus-operators. In a Krein space a densely-defined isometric operator
(i.e.
for all
) 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
,
and
, then
is unitary.
The spectrum of a self-adjoint operator 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 ,
are isolated eigen values of the self-adjoint operator
in a Krein space, then for the corresponding Riesz projections
,
one has
, and if, e.g.,
, then the restrictions
and
have the same Jordan structure. If in a
-space the symmetric operator
has a real non-semi-simple eigen value
, then the corresponding algebraic eigen space
can be decomposed into a direct orthogonal sum:
, where
is a positive subspace contained in the geometric eigen space of
at
, and
is invariant under
with
; if
are the lengths of the Jordan chains of
, one puts
; if
is a non-real eigen value of
, one defines
as the dimension of the corresponding algebraic eigen space. Then
, where the sum extends over all eigen values
of
in the closed upper half-plane. In particular, the length of any Jordan chain of
is
, and the number of eigen values of
in the open upper half-plane, and also the number of non-semi-simple eigen values of
, does not exceed
.
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 -space has a
-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
in a Krein space has a maximal non-negative invariant subspace if
is compact and, additionally,
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
, is to establish the existence of a fixed point
of the fractional-linear transformation
![]() |
where is a contraction from
into
(an angular operator) and
is the matrix representation of
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)
is unitary and
is uniformly bounded for all
; 2)
for all
,
, and
; and 3)
is bounded, self-adjoint and there exists a polynomial
such that
(
). In many cases these maximal non-positive invariant subspaces
can be specified by properties of the spectrum of
. E.g., if
is bounded, self-adjoint and
is compact, then
can be chosen such that
. 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
-space has a common maximal non-negative invariant subspace (M.A. Naimark; for applications in the representation theory of groups in
-spaces see [a19]). Phillips asked ([a16]) if a dual pair of subspaces of
which are invariant under a commutative algebra
of bounded self-adjoint operators in the Krein space
can always be extended to a maximal dual pair whose subspaces are still invariant under
(which would imply that each bounded self-adjoint operator in
has a maximal non-negative invariant subspace). Only partial solutions to this problem are known (cf. [a4], [a2], [a14]).
A self-adjoint operator in the Krein space
is called definitizable (positizable in [a4]) if
and if there exists a polynomial
such that
(
). Each self-adjoint operator
in a
-space has this property (where
can be chosen to be
with
the minimal polynomial of
,
being a
-dimensional non-positive invariant subspace of
); also, each self-adjoint operator
in a Krein space for which
and for which the Hermitian sesquilinear form
(
) has a finite number of negative squares, is definitizable.
The non-real spectrum of the definitizable operator
consists of at most finitely many eigen values, and
has a spectral function, with possibly certain critical points [a13], [a2]. This means that there is a finite set
(of critical points) such that on the semi-ring
, consisting of all bounded intervals of
with end points not in
and their complements, a homomorphism
with values in the set of all self-adjoint projections in the Krein space
is defined, such that for
: a)
is a positive (negative) subspace if
(respectively,
) on
for some definitizing polynomial
of
; b)
is in the double commutant of the resolvent of
; and c) if
is bounded, then
and
,
. If, in particular,
is bounded and
(
), then
, and one has
![]() |
for some bounded operator such that
,
,
(
).
If the spectrum of a definitizable operator is discrete, then the linear span of its algebraic eigen spaces is dense in
; if
is compact and self-adjoint in a
-space
and
, then there is a Riesz basis of
consisting of eigen and associated vectors of
(I.S. Iokhvidov).
There is a theory of extensions of symmetric operators to self-adjoint operators and of generalized resolvents in -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
in a Krein space
has a unitary dilation
in some Krein space
([a2]). In this context one has the following result: Let
,
be Krein spaces,
a simply-connected open domain with smooth boundary such that
,
, and let
be a function which is holomorphic in
whose values are bounded linear operators from
to
. Then there exists a Krein space
and a unitary operator
![]() |
such that
![]() |
(T.Ya. Azizov, see [a2], [a6]; here unitary means that maps the Krein space
continuously onto the Krein space
, 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 on
, where
,
are
matrices,
,
, and let
be the corresponding matrizant (cf. Cauchy operator):
,
. Then
is
-unitary (that is, unitary with respect to the inner product defined in
by the matrix
, see (a3)), and, e.g., in the stability theory for periodic equations
the classification of the eigen values of
into those of positive or negative type plays an essential role ([a5], [a8]). II) The integral operator
,
real and of bounded variation on the interval
,
(
), is self-adjoint in the Krein space
. 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
,
bounded self-adjoint operators in some Hilbert space
, one can associate the so-called companion operator
![]() |
which is self-adjoint in the Krein space ,
(
), where
is the inner product in
and
![]() |
If, e.g., and
is compact and
, the results about the existence of maximal non-negative invariant subspaces mentioned above imply that there exists a bounded linear operator
in
satisfying
,
and
[a12]. In a similar way, if
,
and
are
matrices such that
has
positive and
negative eigen values, the solutions
of the matrix Riccati equation
![]() |
with are in bijective correspondence with all maximal non-positive subspaces which are invariant under the self-adjoint operator
in the
-dimensional Krein space
, equipped with the indefinite inner product (a4) (see [a8]). V) If
is a formally-symmetric regular ordinary differential operator on the interval
with symmetric boundary conditions at
and
, and
is a summable function on
which is not of constant sign (a.e.) on
, then the differential equation
leads to a self-adjoint operator
in the Krein space
with inner product
. If
is semi-bounded from below, the operator
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
the problem
![]() |
which is supposed to have a limit point at and with a boundary condition
at
(
,
are functions which are holomorphic on some set
and satisfying a symmetry condition). The solution of this problem can be represented as
(
), where, in general,
is a self-adjoint operator in some Krein space
and
is the orthogonal projection from
onto
[a17]. VII) Certain classes of analytic functions are closely related to the theory of operators in
-spaces. This concerns, e.g., functions
which are defined and meromorphic in the upper half-plane (or the unit disc) and which are such that the kernel
![]() |
(or
![]() |
has negative squares (that is, for arbitrary
and
, the matrix
has at most
negative eigen values and for at least one choice of
it has
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
-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 ![]() |
[a12] | M.G. Krein, H. Langer, "Ueber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume ![]() |
[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 ![]() |
[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 |
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