Exponentially dominated infinite block matrices of finite Kronecker rank

A. Ben-Artzi; I. Gohberg; M. A. Kaashoek

doi:10.1007/BF01225212

Exponentially dominated infinite block matrices of finite Kronecker rank

A. Ben-Artzi^*, I. Gohberg, M. A. Kaashoek

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

This paper introduces a new class (denoted by EK) of operators acting on a direct sum of m copies of ℓ². Inversion and Fredholm properties of operators in EK are studied. The class EK includes block Toeplitz operators with a rational matrix symbol and non-Toeplitz operators like operators defined by infinite block band matrices. Operators in EK are characterized by the following two properties: (1) the entries a_jk in their canonical block matrix representation decay exponentially as functions of |j-k|, and (2) the Kronecker rank is finite. The main tool in the analysis is based on the fact that operators in the class EK can be represented as input-output operators of certain finite dimensional linear time-variant input-output systems. In the description and the study of these systems dichotomy of difference equations plays an important role.

Original language	English
Pages (from-to)	30-77
Number of pages	48
Journal	Integral Equations and Operator Theory
Volume	18
Issue number	1
DOIs	https://doi.org/10.1007/BF01225212
State	Published - Mar 1994

Keywords

AMS classification: Primary 47A53, 93A25, Secondary 93B15, 39A11

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10.1007/BF01225212

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abstract = "This paper introduces a new class (denoted by EK) of operators acting on a direct sum of m copies of ℓ2. Inversion and Fredholm properties of operators in EK are studied. The class EK includes block Toeplitz operators with a rational matrix symbol and non-Toeplitz operators like operators defined by infinite block band matrices. Operators in EK are characterized by the following two properties: (1) the entries ajk in their canonical block matrix representation decay exponentially as functions of |j-k|, and (2) the Kronecker rank is finite. The main tool in the analysis is based on the fact that operators in the class EK can be represented as input-output operators of certain finite dimensional linear time-variant input-output systems. In the description and the study of these systems dichotomy of difference equations plays an important role.",

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year = "1994",

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AU - Ben-Artzi, A.

AU - Gohberg, I.

AU - Kaashoek, M. A.

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N2 - This paper introduces a new class (denoted by EK) of operators acting on a direct sum of m copies of ℓ2. Inversion and Fredholm properties of operators in EK are studied. The class EK includes block Toeplitz operators with a rational matrix symbol and non-Toeplitz operators like operators defined by infinite block band matrices. Operators in EK are characterized by the following two properties: (1) the entries ajk in their canonical block matrix representation decay exponentially as functions of |j-k|, and (2) the Kronecker rank is finite. The main tool in the analysis is based on the fact that operators in the class EK can be represented as input-output operators of certain finite dimensional linear time-variant input-output systems. In the description and the study of these systems dichotomy of difference equations plays an important role.

AB - This paper introduces a new class (denoted by EK) of operators acting on a direct sum of m copies of ℓ2. Inversion and Fredholm properties of operators in EK are studied. The class EK includes block Toeplitz operators with a rational matrix symbol and non-Toeplitz operators like operators defined by infinite block band matrices. Operators in EK are characterized by the following two properties: (1) the entries ajk in their canonical block matrix representation decay exponentially as functions of |j-k|, and (2) the Kronecker rank is finite. The main tool in the analysis is based on the fact that operators in the class EK can be represented as input-output operators of certain finite dimensional linear time-variant input-output systems. In the description and the study of these systems dichotomy of difference equations plays an important role.

KW - AMS classification: Primary 47A53, 93A25, Secondary 93B15, 39A11

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Exponentially dominated infinite block matrices of finite Kronecker rank

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