A note on Weyl-type theorems and restrictions

Abstract

Let $X$ be a Banach space, let $T\in L\left(X\right)$ be a bounded linear operator, and let $T_n$ be a restriction of $T$ on $R\left(T^n\right)$. This article should be viewed as a note on the research work of Carpintero et al. We give here several different proofs for completeness, and we show the relations of $T$ and $T_n$ to a much greater extent. Moreover, we give sufficient conditions for which Weyl-type theorems for $T$ are equivalent to Weyl-type theorems for $T_n$.