Branko Ćurgus

On the Regularity of the Critical Point Infinity of Definitizable Operators

In this note necessary and sufficient conditions for the regularity of the critical point infinity of a definitizable operator A are given. Using these criteria it is proved that the regularity of the critical point infinity is preserved under some additive perturbations as well as for some operators which are related to A. Applications to indefinite Sturm-Liouville problems are indicated.

The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces

The boundary eigenvalue problems for the adjoint of a symmetric relation S in a Hilbert space with finite, not necessarily equal, defect numbers, which are related to the selfadjoint Hilbert space extensions of S are characterized in terms of boundary coefficients and the reproducing kernel Hilbert spaces they induce.

Quasi-Uniformly Positive Operators in Krein Space

Definitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces. A sufficient condition for definitizability of a selfadjoint operator A with a nonempty resolvent set ρ(A) in a Krein space (H,[·❘·]) is the finiteness of the number of negative squares of the form [Ax❘y] (see [10, p. 11]).

Positive Differential Operators in Krein Space L2(ℝ)

Consider the weighted eigenvalue problem

Riesz Bases of Root Vectors of Indefinite Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions, I

Lu = {\rm \lambda }\left( {{\rm sgn}\;{\rm x}} \right)u,

The Riesz Basis Property of an Indefinite Sturm–Liouville Problem with Non-Separated Boundary Conditions

(1)

Form Domains and Eigenfunction Expansions for Differential Equations with Eigenparameter Dependent Boundary Conditions

Certain meromorphic matrix valued functions on C\R; the so-called boundary coefficients, are characterized in terms of a standard symmetric operator S in a Pontryagin space with finite (not necessarily equal) defect numbers, a meromorphic mapping into the defect subspaces of S; and a boundary mapping for S: Under some simple assumptions the boundary coefficients also satisfy a minimality condition. It is shown that these assumptions hold if and only if for S a generalized von Neumann equality is valid. r 2002 Elsevier Science (USA). All rights reserved. MSC: primary 47B50; 47B25; 34B07; 47B32; secondary 46C20; 47A06

A generalization of Routh's triangle theorem

We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigenparameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space.