The Berkani's property and a note on some recent results

Abstract

In this paper, we continue the study of property (U WΠ) introduced in [5], in connection with other Weyl type theorems. Moreover, we give counterexamples to show that some recent results related to this property, which are announced and proved by P. Aiena and M. Kachad in [2] are false. Furthermore, we specify the mistakes committed in each of them and we give the correct versions. We also give a global note on the paper [7]. IntroductionWe continue the study of properties introduced in [4, 5], in connection with other properties and Weyl type theorems. Moreover, we prove by counterexample (see Remark I), that we do not expected that an operator satisfying property (U W Π ) satisfies property (Z Πa ), contrarily to what has been proved in [2, Thorem 2.5]. Furthermore, we give the correct version of [2, Thorem 2.5] by proving in Theorem 3.7, that if T ∈ L(X) is an operator satisfying property (U W Π ) and Π a (T )∩σ uw (T ) = ∅, then it satisfies property (Z Πa ). We also give the correct version of [2, Thorem 2.6] (see Remark II and Theorem 3.12). As deduction, we give analogous result to Theorem 3.12 for the properties (U W E ) and (Z Ea ). And, we give in Remark IV a very crucial observation on the results of the paper [7]. Terminology and preliminariesLet X denote an infinite dimensional complex Banach space, and we denote by L(X) the algebra of all bounded linear operators on X. For T ∈ L(X), we denote by α(T ) the dimension of the kernel N (T ) and by β(T ) the codimension of the range R(T ). By σ(T ) and σ a (T ), we denote the spectrum, the approximate spectrum of T, respectively. For an operator T ∈ L(X), the ascent p(T ) and the descent q(T ) of T are defined by p(T ) = inf{n ∈ N : N (T n ) = N (T n+1 )} and q(T ) = inf{n ∈ N : R(T n ) = R(T n+1 )}, respectively; the infimum over the empty set is taken ∞.In order to simplify, and to give a global view to the reader, we use the same symbols and notations used in [2]. For more details on the several classes and spectra originating from Fredholm theory and B-Fredholm theory, we refer the reader to [2,5,8].