The spectral continuity of essentially hyponormal operators
Commun. Korean Math. Soc. 2014 Vol. 29, No. 3, 401-408
Printed July 1, 2014
An-Hyun Kim and Eun-Jin Ryu
Changwon National University, Changwon National University
Abstract : If $\mathfrak A$ is a unital Banach algebra, then the spectrum can be viewed as a function $\sigma:{\mathfrak A}\rightarrow \mathfrak {S}$, mapping each $T\in\mathfrak A$ to its spectrum $\sigma(T)$, where $\mathfrak{S}$ is the set, equipped with the Hausdorff metric, of all compact subsets of ${\mathbb C}$. This paper is concerned with the continuity of the spectrum $\sigma$ via Browder's theorem. It is shown that $\sigma$ is continuous when $\sigma$ is restricted to the set of essentially hyponormal operators for which Browder's theorem holds, that is, the Weyl spectrum and the Browder spectrum coincide.
Keywords : spectrum, essential spectrum, spectral continuity, Weyl's theorem, Browder's theorem
MSC numbers : 47A10, 47A53, 47B20, 47B35
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