Generalized Kato Decomposition and Essential Spectra

Abstract

Let R denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear operator T on a Banach space X we prove that $$T=T_M\oplus T_N$$ with $$T_M \in $$ R and $$T_N$$ quasinilpotent (nilpotent) if and only if T admits a generalized Kato decomposition (T is of Kato type) and 0 is not an interior point of the corresponding spectrum $$\sigma _\mathbf{R}(T) =\{\lambda \in \mathbb {C}: T-\lambda \notin \mathbf{R}\}$$ . Moreover, we prove that if $$T-\lambda _0$$ admits a generalized Kato decomposition, then $$\sigma _\mathbf{R}(T)$$ does not cluster at $$\lambda _0$$ if and only if $$\lambda _0$$ is not an interior point of $$\sigma _\mathbf{R}(T)$$ . As a consequence we get several results on cluster points of essential spectra. In that way we extend some results regarding the approximate point spectrum and the surjective spectrum given by Aiena and Rosas (J. Math. Anal. Appl. 279:180–188, 2003), as well as results given by Jiang and Zhong (J. Math. Anal. Appl. 356:322–327, 2009) to the cases of essential spectra.