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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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