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Restriction of an operator to the range of its powers

M. Berkani — 2000

Studia Mathematica

Let T be a bounded linear operator acting on a Banach space X. For each integer n, define T n to be the restriction of T to R ( T n ) viewed as a map from R ( T n ) into R ( T n ) . In [1] and [2] we have characterized operators T such that for a given integer n, the operator T n is a Fredholm or a semi-Fredholm operator. We continue those investigations and we study the cases where T n belongs to a given regularity in the sense defined by Kordula and Müller in[10]. We also consider the regularity of operators with topological...

An Atkinson-type theorem for B-Fredholm operators

M. BerkaniM. Sarih — 2001

Studia Mathematica

Let X be a Banach space and let T be a bounded linear operator acting on X. Atkinson's well known theorem says that T is a Fredholm operator if and only if its projection in the algebra L(X)/F₀(X) is invertible, where F₀(X) is the ideal of finite rank operators in the algebra L(X) of bounded linear operators acting on X. In the main result of this paper we establish an Atkinson-type theorem for B-Fredholm operators. More precisely we prove that T is a B-Fredholm operator if and only if its projection...

Extended Weyl type theorems

M. BerkaniH. Zariouh — 2009

Mathematica Bohemica

An operator T acting on a Banach space X possesses property ( gw ) if σ a ( T ) σ SBF + - ( T ) = E ( T ) , where σ a ( T ) is the approximate point spectrum of T , σ SBF + - ( T ) is the essential semi-B-Fredholm spectrum of T and E ( T ) is the set of all isolated eigenvalues of T . In this paper we introduce and study two new properties ( b ) and ( gb ) in connection with Weyl type theorems, which are analogous respectively to Browder’s theorem and generalized Browder’s theorem. Among other, we prove that if T is a bounded linear operator acting on a Banach space X , then...

A note on the index of B -Fredholm operators

M. BerkaniDagmar Medková — 2004

Mathematica Bohemica

From Corollary 3.5 in [Berkani, M; Sarih, M.; Studia Math. 148 (2001), 251–257] we know that if S , T are commuting B -Fredholm operators acting on a Banach space X , then S T is a B -Fredholm operator. In this note we show that in general we do not have error ( S T ) = error ( S ) + error ( T ) , contrarily to what has been announced in Theorem 3.2 in [Berkani, M; Proc. Amer. Math. Soc. 130 (2002), 1717–1723]. However, if there exist U , V L ( X ) such that S , T , U , V are commuting and U S + V T = I , then error ( S T ) = error ( S ) + error ( T ) , where error stands for the index of a B -Fredholm operator.

Single valued extension property and generalized Weyl’s theorem

M. BerkaniN. CastroS. V. Djordjević — 2006

Mathematica Bohemica

Let T be an operator acting on a Banach space X , let σ ( T ) and σ B W ( T ) be respectively the spectrum and the B-Weyl spectrum of T . We say that T satisfies the generalized Weyl’s theorem if σ B W ( T ) = σ ( T ) E ( T ) , where E ( T ) is the set of all isolated eigenvalues of T . The first goal of this paper is to show that if T is an operator of topological uniform descent and 0 is an accumulation point of the point spectrum of T , then T does not have the single valued extension property at 0 , extending an earlier result of J. K. Finch and a...

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