Abstract
This paper introduces a new class (denoted byEK) of operators acting on a direct sum ofm copies of ℓ2. Inversion and Fredholm properties of operators inEK are studied. The classEK includes block Toeplitz operators with a rational matrix symbol and non-Toeplitz operators like operators defined by infinite block band matrices. Operators inEK are characterized by the following two properties: (1) the entriesa 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 classEK 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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Ben-Artzi, A., Gohberg, I. & Kaashoek, M.A. Exponentially dominated infinite block matrices of finite Kronecker rank. Integr equ oper theory 18, 30–77 (1994). https://doi.org/10.1007/BF01225212
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DOI: https://doi.org/10.1007/BF01225212