Open Access Published by De Gruyter Open Access

Volume 1 Issue 2013 -

January 2013

Issue of Concrete Operators

Contents
Journal Overview

Open Access October 23, 2012

An overview of some recent developments on the Invariant Subspace Problem

Isabelle Chalendar, Jonathan R. Partington

Page range: 1-10

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Abstract

An overview of some recent developments on the Invariant Subspace Problem This paper presents an account of some recent approaches to the Invariant Subspace Problem. It contains a brief historical account of the problem, and some more detailed discussions of specific topics, namely, universal operators, the Bishop operators, and Read’s Banach space counter-example involving a finitely strictly singular operator.

Open Access October 23, 2012

On the Normality of the Unbounded Product of Two Normal Operators

Mohammed Hichem Mortad

Page range: 11-18

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Abstract

On the Normality of the Unbounded Product of Two Normal Operators Let A and B be two -non necessarily bounded- normal operators. We give new conditions making their product normal. We also generalize a result by Deutsch et al on normal products of matrices.

Open Access July 29, 2013

On extended eigenvalues and extended eigenvectors of truncated shift

Hasan Alkanjo

Page range: 19-27

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Abstract

In this paper we consider the truncated shift operator S u on the model space K 2 u := H 2 θ uH 2 . We say that a complex number λ is an extended eigenvalue of Su if there exists a nonzero operator X, called extended eigenvector associated to λ, and satisfying the equation S u X = λXS u . We give a complete description of the set of extended eigenvectors of S u , in the case of u is a Blaschke product..

Open Access September 16, 2013

A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

A. Perälä, J. A. Virtanen, L. Wolf

Page range: 28-36

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Abstract

We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

Open Access October 17, 2013

Factorization of rational matrix functions and difference equations

J.S. Rodríguez, L.F. Campos

Page range: 37-53

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Abstract

In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix function, relative to the unit circle. The constructed method is based on the relation between the general solution of a homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients.

About this journal

Concrete Operators is a fully peer-reviewed open access journal publishing articles devoted to all special classes of linear operators acting on function spaces. This includes, but is not limited to, Perron-Frobenius operators, multiplication operators, operators in reproducing Kernel Hilbert spaces, weighted composition operators, integral operators, kernel operators, Toeplitz operators, Hankel operators, Wiener-Hopf operators, Jacobi operators, weighted shift operators, operators on function spaces.