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 ⊆ X such that Tn (X) ⊆ W, for some n ≥ 1, where T ∈ 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 semiFredholm 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.
Highlights
IntroductionAfter them many authors have introduced and studied a large number of spectral properties associated to an operator by using spectra derived from either Fredholm operators theory or B-Fredholm operators theory
Today all these results are known as Weyl type theorems or Weyl type properties and over the last years there has been a considerable interest to study these properties in operator theory
In this paper, using the framework dealt by Barnes [6], we study the behavior of certain spectral properties of an operator T on a proper closed and T -invariant subspace W ⊆ X such that T n(X) ⊆ W, for some n ≥ 1, where T is a bounded linear operator acting on an infinite-dimensional complex Banach space X
Summary
After them many authors have introduced and studied a large number of spectral properties associated to an operator by using spectra derived from either Fredholm operators theory or B-Fredholm operators theory. In this paper, using the framework dealt by Barnes [6] (which extends the context treated by Berkani [7]), we study the behavior of certain spectral properties of an operator T on a proper closed and T -invariant subspace W ⊆ X such that T n(X) ⊆ W , for some n ≥ 1, where T is a bounded linear operator acting on an infinite-dimensional complex Banach space X. We prove that for these subspaces (which generalize the case R(T n) closed for some n ≥ 0, [16], [18]) a large number of spectral properties are transmitted from T to its restriction on W and vice-versa. We give some applications of these results for integral operators acting on certain functions spaces
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