Functional models and finite dimensional perturbations of the shift

We study finite dimensional perturbations of shift operators and their membership to the classes A _{m, n} appearing in the theory of dual algebras. The results obtained yield informations about the lattice of invariant subspaces via the techniques of Scott Brown.     

On two-sided interpolation for upper triangular Hilbert-Schmidt operators

Motivated by the theory of nonstationary linear systems a number of problems in the theory of analytic functions have analogues in the setting of upper-triangular operators, where the complex variable is replaced by a diagonal operator. In this paper we focus on the analogue of interpolation in the Hardy space H₂ and study a two-sided Nudelman type interpolation problem in the framework of upper-triangular Hilbert-Schmidt operators.     

Compact linear operators on functional spaces with two norms

Some principles of the operator theory in a linear space with two norms are established in this paper. The well-known Hilbert-Schmidt theorem on the eigenfunction expansion of sourcewise represented functions, Mercer's theorem and other results can be consider as special cases of the statements presented. The general approach proposed is used to construct the theory of symmetrizable operators and to investigate the asymptotic behaviour of eigenvalues of compact operators.This paper was translated by M. Gorbuchuk and V. GorbachukThis paper was translated by M. Gorbuchuk and V. Gorbachuk.     

Commutant lifting theorem and interpolation in discrete nest algebras

Ball in [Ba] showed that the commutant lifting theorem for the nest algebras due to Paulsen and Power gives a unified approach to a wide class of interpolation problems for nest algebras. By restricting our attention to the case when nest algebras associated with the problems are discrete we derive a variant of the commutant lifting theorem which avoids language of representation theory and which is sufficient to treat an analog of the generalized Schur-Nevannlinna-Pick (SNP) problem in the setting of upper triangular operators.     

Model theory for hyponormal contractions

Agler's abstract model theory is applied to the family of hyponormal contractions. A sufficient condition for an operator to be extremal in this family is given, and this is used to show that the boundary, or smallest model, for the family is the whole family.     

Decompositions of Singular Continuous Spectra of \mathcal{H}_{ - 2} -class Rank One Perturbations

The decomposition theory for the singular continuous spectrum of rank one singular perturbations is studied. A generalization of the well-known Aronszajn-Donoghue theory to the case of decompositions with respect to α-dimensional Hausdorff measures is given and a characterization of the supports of the α-singular, α-absolutely continuous, and strongly α-continuous parts of the spectral measure of - class rank one singular perturbations is given in terms of the limiting behaviour of the regularized Borel transform.     

Finite Laurent Developments and the Logarithmic Residue Theorem in the Real Non-analytic Case

This paper develops a general abstract non-holomorphic operator calculus under minimal regularity requirements on the family of operators through the concept of algebraic eigenvalue and the use of a, very recent, transversalization theory. Further, it analyzes under what conditions the inverse of a non-analytic family admits a finite Laurent development, and employs the new findings to calculate the multiplicity of a real non-analytic family through a logarithmic residue, so extending the applicability of the classical theory of I. C. Gohberg and coworkers. Applications to matrix families and Nonlinear Analysis are also explained.     

SOME SPECTRAL PROPERTIES OF MULTIPLIERS ON SEMI-PRIME BANACH ALGEBRAS

Abstract

We extend to arbitrary semi-prime Banach algebras some results of spectral theory and Fredholm theory obtained in [1] and [2] for multipliers defined in commutative semi-simple Banach algebras.     

Q-functions of Hermitian Contractions of Krein-Ovcharenko Type

In this paper operator-valued Q-functions of Krein-Ovcharenko type are introduced. Such functions arise from the extension theory of Hermitian contractive operators A in a Hilbert space ℌ. The definition is related to the investigations of M.G. Krein and I.E. Ovcharenko of the so-called Q_μ- and Q_M-functions. It turns out that their characterizations of such functions hold true only in the matrix valued case. The present paper extends the corresponding properties for wider classes of selfadjoint contractive extensions of A. For this purpose some peculiar but fundamental properties on the behaviour of operator ranges of positive operators will be used. Also proper characterizations for Q_μ- and Q_M-functions in the general operator-valued case are given. Shorted operators and parallel sums of positive operators will be needed to give a geometric understanding of the function-theoretic properties of the corresponding Q-functions.     

On semibounded restrictions of self-adjoint operators

After the von Neumann's remark [10] about pathologies of unbounded symmetric operators and an abstract theorem about stability domain [9], we develope here a general theory allowing to construct semibounded restrictions of selfadjoint operators in explicit form. We apply this theory to quantum-mechanical momentum (position) operator to describe corresponding stability domains. Generalization to the case of measurable functions of these operators is considered. In conclusion we discuss spectral properties of self-adjoint extensions of constructed self-adjoint restrictions.     

A note on operator probability theory involving numerical ranges

In this note, the relationships between the expectation and variance in operator probability theory and numerical range of operators are considered.     

Bitangential structured interpolation theory

In this paper we present a generalization of the classical bitangential Nevanlinna-Pick theory in which one bounds not the norm of the interpolating functions but their structured singular value This work was motivated by some problems arising in robust control of systems with structured uncertainty. This approach is based on the commutant lifting theory of Sz.-Nagy and Foias (1968) and extends previous work of Bercovici, Foias and Tannenbaum (1990) on structured matrix Nevanlinna-Pick interpolation.This work was supported in part by grants from the National Science Foundation DMS-8811084, ECS-9122106, by the Air Force Office of Scientific Research F49620-94-1-0058DEF and by the Army Research Office DAAL03-91-G-0019, DAAH04-93-G-0332.     

Joint seminormality and Dirac operators

In this article, an approach to joint seminormality based on the theory of Dirac and Laplace operators on Dirac vector bundles is presented. To eachn-tuple of bounded linear operators on a complex Hilbert space we first associate a Dirac bundle furnished with a metric-preserving linear connection defined in terms of thatn-tuple. Employing standard spin geometry techniques we next get a Bochner type and two Bochner-Kodaira type identities in multivariable operator theory. Further, four different classes of jointly seminormal tuples are introduced by imposing semidefiniteness conditions on the remainders in the corresponding Bochner-Kodaira identities. Thus we create a setting in which the classical Bochner's method can be put into action. In effect, we derive some vanishing theorems regarding various spectral sets associated with commuting tuples. In the last part of this article we investigate a rather general concept of seminormality for self-adjoint tuples with an even or odd number of entries.     

Spectral analysis of the Direct Sum Hamiltonian Operators

In this paper we investigate the deficiency indices theory and the selfad-joint and nonselfadjoint (dissipative, accumulative) extensions of the minimal symmetric direct sum Hamiltonian operators. In particular using the equivalence of the Lax-Phillips scattering matrix and the Sz.-Nagy-Foia¸s characteristic function, we prove that all root (eigen and associated) vectors of the maximal dissipative extensions of the minimal symmetric direct sum Hamiltonian operators are complete in the Hilbert spaces.     

m-Isometric Commuting Tuples of Operators on a Hilbert Space

We consider a generalization of isometric Hilbert space operators to the multivariable setting. We study some of the basic properties of these tuples of commuting operators and we explore several examples. In particular, we show that the d-shift, which is important in the dilation theory of d-contractions (or row contractions), is a d-isometry. As an application of our techniques we prove a theorem about cyclic vectors in certain spaces of analytic functions that are properly contained in the Hardy space of the unit ball of .     

Scattering Matrix,Phase Shift,Spectral Shift and Trace Formula for One-dimensional Dissipative Schrödinger-type Operators

The paper is devoted to Schr?dinger operators with dissipative boundary conditions on bounded intervals. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed. In particular, the trace formula and the Birman-Krein formula are verified directly. The results are exploited for dissipative Schr?dinger-Poisson systems. In friendship dedicated to P. Exner on the occasion of his 60^th birthday This work was supported by DFG, Grant 1480/2.     

Double Operator Integrals in a Hilbert Space

Double operator integrals are a convenient tool in many problems arising in the theory of self-adjoint operators, especially in the perturbation theory. They allow to give a precise meaning to operations with functions of two ordered operator-valued non-commuting arguments. In a different language, the theory of double operator integrals turns into the problem of scalarvalued multipliers for operator-valued kernels of integral operators.The paper gives a short survey of the main ideas, technical tools and results of the theory. Proofs are given only in the rare occasions, usually they are replaced by references to the original papers. Various applications are discussed.     

Relaxation of metric constrained interpolation and a new lifting theorem

In this paper a new lifting interpolation problem is introduced and an explicit solution is given. The result includes the commutant lifting theorem as well as its generalizations in [27] and [2]. The main theorem yields explicit solutions to new natural variants of most of the metric constrained interpolation problems treated in [9]. It is also shown that via an infinite dimensional enlargement of the underlying geometric structure a solution of the new lifting problem can be obtained from the commutant lifting theorem. However, the new setup presented obtained from the commutant lifting theorem. However, the new setup presented in this paper appears to be better suited to deal with interpolations problems from systems and control theory than the commutant lifting theorem.Dedicated to Israel Gohberg, as a token of admiration for his inspiring work in analysis and operator theory, with its far reaching applications, in friendship and with great affection.     

Defect operators, defect functions and defect indices for analytic submodules

This paper mainly concerns defect operators and defect functions of Hardy submodules, Bergman submodules over the unit ball, and Hardy submodules over the polydisk. The defect operator (function) carries key information about operator theory (function theory) and structure of analytic submodules. The problem when a submodule has finite defect is attacked for both Hardy submodules and Bergman submodules. Our interest will be in submodules generated by polynomials. The reason for choosing such submodules is to understand the interaction of operator theory, function theory and algebraic geometry.