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A characterization of Riesz operators. (English) Zbl 0618.47014
Let \({\mathcal A}(E)\) be the class of all bounded linear operators on a complex Banach space E having the property that the restriction of A on any closed infinite-dimensional invariant subspace M for A is not bijective. A bounded operator A on E is said to be a Riesz operator if for each complex \(\lambda\) \(\neq 0\), \(\lambda\) I-A is a Fredholm operator. In this note we give the following characterization of the class \({\mathcal R}(E)\) of all Riesz operator.
Theorem: \(A\in {\mathcal R}(E)\) if and only if the following conditions hold
a) \(A\in {\mathcal A}(E).\)
b) For each spectral point \(\lambda\) \(\neq 0\) there exists a spectral set \(\sigma\) which contains \(\lambda\) such that \(0\not\in \sigma.\)
An example of a convolution operator \(T_{\mu}\in {\mathcal A}(E)\) defined on the group algebra \(E=L_ 1(G)\), where G is a compact Abelian group, shows that the inclusion \({\mathcal A}(E)\supseteq {\mathcal R}(E)\) is generally proper. Moreover the class \({\mathcal A}(E)\) may be replaced with the class \({\mathcal A}_ 0(E)\) of all bounded linear operators A having the property that the restriction of A on any closed infinite-dimensional invariant subspace M for A does not admit a bounded inverse.

MSC:
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
47A53 (Semi-) Fredholm operators; index theories
47A25 Spectral sets of linear operators
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