Communications of the
Korean Mathematical Society CKMS

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