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Similarity of a linear strict set-contraction and the radius of the essential spectrum


Author: Mau-Hsiang Shih
Journal: Proc. Amer. Math. Soc. 100 (1987), 137-139
MSC: Primary 47A65
MathSciNet review: 883416
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Abstract: If $ A$ is a bounded linear operator on a Hilbert space, define $ {r_e}(A)$, the essential spectral radius of $ A$, by

$\displaystyle {r_e}(A): = \sup \{ \vert\lambda \vert:\lambda \in \operatorname{ess}(A) = \operatorname{essential}\;\operatorname{spectrum}\;{\text{of}}\;A\} .$

It is shown that

$\displaystyle {r_e}(A) = \operatorname{inf}\{ \alpha ({S^{ - 1}}AS)\vert S:H \t... ...;{\text{bounded}}\;{\text{invertible}}\;{\text{linear}}\;\operatorname{map}\} ,$

where $ \alpha $ is the Kuratowski measure of noncompactness. As a consequence, a charcterization of the similarity of a linear strict set-contraction is obtained.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0883416-2
Keywords: Essential spectrum, $ k$-set-contraction, measure of noncompactness, similarity of operators
Article copyright: © Copyright 1987 American Mathematical Society