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On the hereditary character of certain spectral properties and some applications

Abstract:

In this paper we study the behavior of certain spectral properties of an operator T on a proper closed and T-invariant subspace W (Formula presented) X such that Tn(X) (Formula presented) W, for some n > 1, where T (Formula presented) L(X) and X is an infinite-dimensional complex Banach space. We prove that for these subspaces a large number of spectral properties are transmitted from T to its restriction on W and vice-versa. As consequence of our results, we give conditions for which semi-Fredholm spectral properties, as well as Weyl type theorems, are equivalent for two given operators. Additionally, we give conditions under which an operator acting on a subspace can be extended on the entire space preserving the Weyl type theorems. In particular, we give some applications of these results for integral operators acting on certain functions spaces.