# 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 -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 |

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Krein space.

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