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 ST is a B-Fredholm operator. In this note we show that in general we do not have ind(ST) = ind(S) + ind(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 US + V T = I, then ind(ST) = ind(S) + ind(T), where ind stands for the index of a B-Fredholm operator.
An operator T acting on a Banach space X possesses property (gw) if σa(T) \ σSBF− + (T) = E(T), where σa(T) is the approximate point spectrum of T, σSBF− + (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 property (gw) holds for T if and only if property (gb) holds for T and E(T) = Π(T), where Π(T) is the set of all poles of the resolvent of T.
Let T be an operator acting on a Banach space X, let σ(T) and σBW (T) be respectively the spectrum and the B-Weyl spectrum of T. We say that T satisfies the generalized Weyl’s theorem if σBW (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 recent result of Aiena and Monsalve. Our second goal is to give necessary and sufficient conditions under which an operator having the single valued extension property satisfies the generalized Weyl’s theorem.