Compact perturbations of normal operators in a Krein space

We prove the existence of the spectral function of, in general, an unbounded normal operator in a Krein space that is the perturbation of a fundamentally reducible normal operator N by an operator of a special class that contain N-compact operators in it.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1299–1306, October, 1990.     

Nonnegative perturbations of selfadjoint operators

Let 4 be a selfadjoint operator on a Hilbert space H. The results in this paper provide necessary and sufficient conditions on A in order that there exist a nontrivial nonnegative operator D and a unitary operator U with UA = (A − D)U. In one case considered, it is required that the least subspace reducing A, U and containing the range of D is the full Hilbert space. In this case the operators U, D exist if and only if the operator A is not a scalar multiple of the identity and the maximum and minimum of the spectrum of A are not eigenvalues of finite multip icity. This result is used to complete a characterization of the absolute value of a completely nonnormal hyponormal operator.     

Smooth rank one perturbations of selfadjoint operators

Let be a selfadjoint operator in a Hilbert space with inner product . The rank one perturbations of have the form , , for some element . In this paper we consider smooth perturbations, i.e. we consider for some . Function-theoretic properties of their so-called -functions and operator-theoretic consequences will be studied.

     

On compact perturbations of finite-zone Jacobi operators

For a bounded Jacobi operator (a discrete analog of the Sturm–Liouville operator on the half-axis), the compactness of a perturbation is studied. The perturbation is produced by a change of the spectral measure (the essential spectrum remains unchanged). Bibliography: 21 titles.     

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 invariant maximal non-negative subspaces of certain operators acting in a Krein space

The aim of this note is to show that a series of proofs for the existence of a maximal non-negative subspace which is invariant under an operator S in a Krein space, or for statements equivalent with this, follows a general pattern, using an approximating net S⁽ⁱ⁾ for S such that for S⁽ⁱ⁾ the existence of such a space is known.     

Definitizable extensions of positive symmetric operators in a Krein space

The Friedrichs extension and the Krein extension of a positive operator in a Krein space are characterized in terms of their spectral functions in a Krein space.     

On the instability of non-semi-Fredholm operators under compact perturbations

The stability of several natural sets of the non-semi-Fredholm operators in a separable Hilbert space under compact perturbations studied by R. Bouldin. (The instability of non-semi-Fredholm operators under compact perturbations, J. Math. Anal. Appl.87 (1982), 632–638.) The aim of the present article is to study this problem in arbitrary Banach spaces. We also derive a curious characterization of separable Banach spaces.     

On the eigenvalues of compact operators in a Hilbert space

Summary This paper contains several remarks concerning the behavior of the eigenvalues of compact operators in a Hilbert space. The results are related to the theorem for a finite matrix obtained as Theorem 1 in my previous paper[5].This work was partially supported by a research grant of the Sakkokai Foundation.     

On commuting compact self-adjoint operators on a Pontryagin space

Suppose thatA ₁,A ₂, ...,A _n are compact commuting self-adjoint linear maps on a Pontryagin spaceK of indexk and that their joint root subspaceM ₀ at the zero eigenvalue in ⁿ is a nondegenerate subspace. Then there exist joint invariant subspacesH andF inK such thatK=FH,H is a Hilbert space andF is finite-dimensional space withkdimF(n+2)k. We also consider the structure of restrictionsA _j|F in the casek=1.     

A class of linear kinetic equations in a Krein space setting

Krein space methods are used to derive the unique solvability of a class of abstract kinetic equations on a half-space with accretive collision operators. At the same time a new proof is provided for the case of a positive self-adjoint collision operator. A Fokker-Planck type example is worked out as a new application.This work was performed while the author resided at the Dept. of Physics and Astronomy of the Free University of Amsterdam, The Netherlands.     

The instability of non-semi-Fredholm operators under compact perturbations

Methods are presented for calculating the perturbation series for the roots of polynomials which contain parameters in addition to the perturbation parameter. In particular the complicating effects of multiplicities of the unperturbed roots are examined.     

On strongly stable normal operators in Krein spaces

For bounded normal operators in Krein spaces we give a necessary and sufficient condition for strong stability. The same result for unitary operators was obtained by M.G.Krein [1] (see also [2]). For selfadjoint operators we refer to the papers of P.Jonas, H.Langer [3] and H.Langer [4].     

On a class of exponential-type operators and their limit semigroups

The paper is mainly focused upon the study of a class of second order degenerate elliptic operators on unbounded intervals.We show that these operators generate strongly continuous semigroups in suitable weighted spaces of continuous functions.Furthermore, we represent the semigroups as limits of iterates of the so-called exponential-type operators.In a particular case, starting from the stochastic differential equations associated with these operators, we also find an integral representation of the semigroup and determine its asymptotic behaviour.