摘要

Let X be a Banach space, let T is an element of L(X) be a bounded linear operator, and let T-n be a restriction of T on R(T-n). 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.