Normal extensions of semigroups of operators to Krein spaces

In this paper, we prove that every strongly continuous semigroup of bounded operators on a Hilbert space may be extended to a strongly continous semigroup of normal operators on a larger Krein space. Several equivalent formulations for the case where the extension space is a Pontrjagin space are given.     

Unitary extensions of two-parameter local semigroups of isometric operators and the Krein extension theorem

A notion of two-parameter local semigroups of isometric operators in Hilbert space is discussed. It is shown that under certain conditions such a semigroup can be extended to a strongly continuous two-parameter group of unitary operators in a larger Hilbert space. As an application a simple proof of the Eskin bidimensional version of the Krein extension theorem is given.     

Invariant Maximal Positive Subspaces and Polar Decompositions

It is proved that invertible operators on a Krein space which have an invariant maximal uniformly positive subspace and map its orthogonal complement into a nonnegative subspace allow polar decompositions with additional spectral properties. As a corollary, several classes of Krein space operators are shown to allow polar decompositions. An example in a finite dimensional Krein space shows that there exist dissipative operators that do not allow polar decompositions.     

Pontryagin spaces of entire functions I

We give a generalization of L.de Branges theory of Hilbert spaces of entire functions to the Pontryagin space setting. The aim of this-first-part is to provide some basic results and to investigate subspaces of Pontryagin spaces of entire functions. Our method makes strong use of L.de Branges's results and of the extension theory of symmetric operators as developed by M.G.Krein.     

Normal Extensions of Operators to Krein Spaces

In this paper, it is proved that every bounded linear operator on a Hilbert space has a normal extension to a Krein space. Two criteria for J-subnormality are given. In particular, in order that T be subnormal, it suffices that exp(-\bar \Lambda T^*)exp(\Lambda T) be a positive definite operator function on a bounded infinite subset of complex plane. This improves the condition of Bram [4]. Also it is proved that the local spectral subspaces are closed for J-subnormal operators.     

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.     

On three Krein extension problems and some generalizations

Three basic extension problems which were initiated by M. G. Krein are discussed and further developed. Connections with interpolation problems in the Carathéodory class are explained. Some tangential and bitangential versions are considered. Full characterizations of the classes of resolvent matrices for these problems are given and formulas for the resolvent matrices of left tangential problems are obtained using reproducing kernel Hilbert space methods.Dedicated to the memory of M. G. Krein, a beacon for us both.The authors wish to acknowledge the partial support of the Israel-Ukraine Exchange Program. D. Z. Arov also wishes to thank the Weizmann Institute of Science for partial support and hospitality; H. Dym wishes to thank Renee and Jay Weiss for endowing the chair which supports his research.     

Some Remarks on Krein's Extension Theory

The extension problem of semibounded symmetric operators and symmetric operators with a gap is studied in detail. Using a suitable representation (Krein model) for the inverses of those operators a parameterization of their symmetric and self-adjoint extensions is introduced which improves Krein's famous extension theory. In particular, the parameterization clearly shows which self-adjoint extensions in the gap case correspond to Friedrichs and v. Neumann or Krein extensions in the semibounded case. Moreover, special properties of the extensions as the exactness of the gap are characterized in terms of the parameters.     

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)     

Dissipative operators in the Krein space. Invariant subspaces and properties of restrictions

We prove that a dissipative operator in the Krein space has a maximal nonnegative invariant subspace provided that the operator admits matrix representation with respect to the canonical decomposition of the space and the upper right operator in this representation is compact relative to the lower right operator. Under the additional assumption that the upper and lower left operators are bounded (the so-called Langer condition), this result was proved (in increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. We relax the Langer condition essentially and prove under the new assumptions that a maximal dissipative operator in the Krein space has a maximal nonnegative invariant subspace such that the spectrum of its restriction to this subspace lies in the left half-plane. Sufficient conditions are found for this restriction to be the generator of a holomorphic semigroup or a C ₀-semigroup.     

Noncommuting Domination in Krein Spaces Via Commutators of Block Operator Matrices

The commutators of 2 × 2 block operator matrices with (unbounded) operator entries are investigated. The matrix representation of a symmetric operator in a Krein space is exploited. As a consequence, the domination result due to Cichoń, Stochel and Szafraniec is extended to the case of Krein spaces.     

Decompositions of a Krein space in regular subspaces invariant under a uniformly bounded C0-semigroup of bi-contractions

We give necessary and sufficient conditions under which a C₀-semigroup of bi-contractions on a Krein space is similar to a semigroup of contractions on a Hilbert space. Under these and additional conditions we obtain direct sum decompositions of the Krein space into invariant regular subspaces and we describe the behavior of the semigroup on each of these summands. In the last section we give sufficient conditions for the co-generator of the semigroup to be power bounded.     

Local spectral theory for normal operators in Krein spaces

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.     

THE ISOMETRIC EXTENSION OF AN INTO MAPPING FROM THE UNIT SPHERE S[e（Γ）] TO THE UNIT SPHERE S（E）

This is such a article to consider an ＂into＂ isometric mapping between two unit spheres of two infinite dimensional spaces of different types. In this article, we find a useful condition （using the Krein-Milman property） under which an into-isometric mapping from the unit sphere of e（Γ） to the unit sphere of a normed space E can be linearly isometric extended.     

Uniqueness of the Continuation of a Certain Function to a Positive Definite Function

Mathematical Notes - In 1940, M. G. Krein obtained necessary and sufficient conditions for the extension of a continuous function f defined in an interval (-a, a), a > 0, to a positive...     

GENERALIZED DILATIONS OF OPERATOR SEQUENCES TO KREIN SPACE

In this paper, the author considers the generalized dilations of operator sequences in a Hilbert space to a Krein space. In order to obtain the unitary and solf-adjoint dilations, only some boundedness and symmetry assumptions are needed.     

The Krein Formula for Generalized Resolvents in Degenerated Inner Product Spaces

 Let S be a symmetric operator with defect index (1,1) in a Pontryagin space ℋ. The Krein formula establishes a bijective correspondence between the generalized resolvents of S and the set of Nevanlinna functions as parameters. We give an analogue of the Krein formula in the case that ℋ is a degenerated inner product space. The set of parameters is determined by a kernel condition. These results are applied to some classical interpolation problems with singular data. Received 3 February 1997; in revised form 9 June 1997     

Description of bilinear forms generated by indefinite vector measures

We describe a correlation function generated by a J-orthogonal indefinite measure with values in a Krein space.