Peter Jonas

On a class of selfadjoint operators in Krein space and their compact perturbations

For a class of selfadjoint operators in a Krein space containing the definitizable selfadjoint operators a funetional calculus and the spectral function are studied. Stability properties of the spectral function with respect to small compact perturbations of the resolvent are proved.

On a Class of Analytic Operator Functions and Their Linearizations

We consider an operator function T in a Krein space which can formally be written as in (0.1) but the last term on the right of (0.1) is replaced by a relatively form-compact perturbation of a similar form. We study relations between the operator function T , a selfadjoint operator M in some Krein space, associated with T , and an operator which can be constructed with the help of the operator function −T −1. The results are applied to a Sturm-Liouville problem with a coefficient depending rationally on the eigenvalue parameter.

Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems

Partial Non-stationary Perturbation Determinants for a Class of J-symmetric Operators.- Reproducing Kernel Spaces of Series of Fueter Polynomials.- Extremal Extensions of a C(?)-suboperator and Their Representations.- A Variational Principle for Linear Pencils of Forms.- Selfadjoint Extensions with Several Gaps: Finite Deficiency Indices.- The Spectrum of the Multiplication Operator Associated with a Family of Operators in a Banach Space.- A Factorization Model for the Generalized Friedrichs Extension in a Pontryagin Space.- Generalized Schur Functions and Augmented Schur Parameters.- On Nonmonic Quadratic Matrix Polynomials with Nonnegative Coefficients.- On Operator Representations of Locally Definitizable Functions.- Symmetric Relations of Finite Negativity.- An Operator-theoretic Approach to a Multiple Point Nevanlinna-Pick Problem for Generalized Caratheodory Functions.- Bounded Normal Operators in Pontryagin Spaces.- Scalar Generalized Nevanlinna Functions: Realizations with Block Operator Matrices.- Polar Decompositions of Normal Operators in Indefinite Inner Product Spaces.- Bounds for Contractive Semigroups and Second-Order Systems.

Selfadjoint Extensions of a Closed Linear Relation of Defect One in a Krein Space

In this paper we study the selfadjoint and the nonnegative selfadjoint extensions of a nonnegative closed linear relation (c.l.r.) A0 of defect one in a Krein space (H, [·, ·]). These extensions are described by their resolvents, that is, M. G. Krein’s formula for the resolvents of the extensions of a symmetric densely defined operator with defect (1,1) is generalized to the situation considered here. The main difficulties which arise with this generalization are the following.

Quadratic (Weakly) Hyperbolic Matrix Polynomials: Direct and Inverse Spectral Problems

Let L be a monic quadratic weakly hyperbolic or hyperbolic n × n matrix polynomial. We solve some direct spectral problems: We prove that the eigenvalues of a compression of L to an (n − 1)-dimensional subspace of ℂ n block-interlace and that the eigenvalues of a one-dimensional perturbation of L (−,+)-interlace the eigenvalues of L. We also solve an inverse spectral problem: We identify two given block-interlacing sets of real numbers as the sets of eigenvalues of L and its compression.

Riggings and relatively form bounded perturbations of nonnegative operators in Kreĭn spaces

For nonnegative operators in Kreĭnn spaces we give conditions for the preservation of the nonemptiness of the resolvent set and the preservation of the regularity of critical points under relatively form bounded perturbations.

Partial Non-stationary Perturbation Determinants for a Class of J-symmetric Operators

We consider the partial non-stationary perturbation determinant

Boundary value problems with local generalized Nevanlinna functions in the boundary condition

\Delta _{H/A}^{(1)} (t): = \det \left( {e^{itA} P_1 e^{ - itH} |_{\mathcal{H}_1 } } \right),t \in \mathbb{R}.