Mark Grigor'evich Krein

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, p. 576, April, 1990.     

Operators in Krein Space

We study self-adjoint operators in Krein space. Our goal is to show that there is a relationship between the following classes of operators: operators with a compact “corner,” definitizable operators, operators of classes (H) and K(H), and operators of class D _κ ⁺.     

Projections in Krein spaces

Let (H,J) be a Krein space with selfadjoint involution J. Starting with a canonical representation of a J-selfadjoint projection, J-projection in short, as the sum of a J-positive projection and a J-negative one we study in detail the structure of a regular subspace, that is, the range of a J-projection. We treat the problem when the sum of two regular subspaces is again regular. We also treat the problem when the closure of the range of the product of a J-contraction and a J-expansion becomes regular.     

Schur Complements in Krein Spaces

The aim of this work is to generalize the notions of Schur complements and shorted operators to Krein spaces. Given a (bounded) J-selfadjoint operator A (with the unique factorization property) acting on a Krein space and a suitable closed subspace of , the Schur complement of A to is defined. The basic properties of are developed and different characterizations are given, most of them resembling those of the shorted of (bounded) positive operators on a Hilbert space. To the memory of Professor Mischa Cotlar     

Normal Projections in Krein Spaces

Given a complex Krein space ${\mathcal{H}}$ with fundamental symmetry J, the aim of this note is to characterize the set of J-normal projections $$\mathcal{Q}=\{Q \in L(\mathcal{H}) : Q^2=Q \,{\rm and}\, Q^{\#}Q=QQ^{\#}\}.$$ The ranges of the projections in ${\mathcal{Q}}$ are exactly those subspaces of ${\mathcal{H}}$ which are pseudo-regular. For a fixed pseudo-regular subspace ${\mathcal{S}}$ , there are infinitely many J-normal projections onto it, unless ${\mathcal{S}}$ is regular. Therefore, most of the material herein is devoted to parametrizing the set of J-normal projections onto a fixed pseudo-regular subspace ${\mathcal{S}}$ .     

Quasinormal operators on Krein space

In this paper we introduce the concept of quasinormal and subnormal operators on a Krein space and prove that every quasinormal operator is subnormal. And some conditions for an operator on a Hilbert space to be a subnormal operator in the Krein space sense are obtained.     

Positive operators on Krein spaces

A Krein operator is a positive operator, acting on a partially ordered Banach space, that carries positive elements to strong units. The purpose of this paper is to present a survey of the remarkable spectral properties (most of which were established by M.G. Krein) of these operators. The proofs presented here seem to be simpler than the ones existing in the literature. Some new results are also obtained. For instance, it is shown that every positive operator on a Krein space which is not a multiple of the identity operator has a nontrivial hyperinvariant subspace. Dedicated to the memory of M.G. Krein (1907–1989)     

Abstract splines in Krein spaces

We present generalizations to Krein spaces of the abstract interpolation and smoothing problems proposed by Atteia in Hilbert spaces: given a Krein space K and Hilbert spaces H and E (bounded) surjective operators T:H→K and V:H→E, ρ>0 and a fixed z₀∈E, we study the existence of solutions of the problems and .     

Krein H* - Triple Systems

In this paper, the authors initiate the study of a class of triple systems called Krein H^*-triple systems and show that a semisimple Krein H^*-triple system is an orthogonal direct sum of simple Krein H^*-triple systems. A separable Krein H_*-triple system satisfying the condition of regularity is the closure of an increasing union of regular Krein H^*-triple systems.This research is supported by the University Grants Commission of India.     

Perturbation of spectra and Krein extensions

Rendiconti del Circolo Matematico di Palermo Series 2 -     

Krein Parameters of Fiber-Commutative Coherent Configurations

For fiber-commutative coherent configurations,we show that Krein parameters can be defined essentially uniquely.As a consequence,the general Krein condition reduces to positive semidefiniteness of finitely many matrices determined by the parameters of a coherent configuration.We mention its implications in the coherent configuration defined by a generalized quadrangle.We also simplify the absolute bound using the matrices of Krein parameters.