Journal of Operator Theory

Volume 57, Issue 2, Spring 2007 pp. 325-346.

Banach algebras of operator sequences: Approximation numbers

Authors: A. Rogozhin (1) and B. Silbermann (2)
Author institution: (1) Department of Mathematics, Chemnitz University of Technology, Chemnitz, 09107, Germany
(2) Department of Mathematics, Chemnitz University of Technology, Chemnitz, 09107, Germany

Summary: In this paper we discuss the asymptotic behavior of the approximation numbers for operator sequences belonging to a special class of Banach algebras. Associating with every operator sequence $\{A_n\}$ from such a Banach algebra a collection $\{W^t\{A_n\}\}_{t \in T}$ of bounded linear operators on Banach spaces $\{\mathbb{E}^t\}_{t \in T},$ i.e.\ $W^t\{A_n\} \in \mathcal{L}(\mathbb{E}^t),$ we establish several properties of approximation numbers of $A_n,$ among them the so-called $k$-splitting property, and show that the behavior of approximation numbers of $A_n$ depends heavily on the Fredholm properties of operators $W^t\{A_n\}.$

Keywords: Approximation numbers, operator sequences, $k$-splitting property.

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