Mediterr. j. math. 5 (2008), 371–378 1660-5446/030371-8, DOI 10.1007/s00009-008-0156-z c  2008 Birkh¨auser Verlag Basel/Switzerland Mediterranean Journal of Mathematics On the Property (gw) Mohammed Amouch and Mohammed Berkani ∗ Abstract. In this note we introduce and study the property (gw), which ex- tends property (w) introduced by Rako˘ cevic in [23]. We investigate the prop- erty (gw) in connection with Weyl type theorems. We show that if T is a bounded linear operator T acting on a Banach space X, then property (gw) holds for T if and only if property (w) holds for T and Πa(T )= E(T ), where Πa(T ) is the set of left poles of T and E(T ) is the set of isolated eigenvalues of T. We also study the property (gw) for operators satisfying the single valued extension property (SVEP). Classes of operators are considered as illustrating examples. Mathematics Subject Classification (2000). 47A53, 47A10, 47A11. Keywords. B-Fredholm operator, Weyl’s theorem, generalized Weyl’s theo- rem, a-Weyl’s theorem, generalized a-Weyl’s theorem, property (gw). 1. Introduction Throughout this paper L(X ) denotes the Banach algebra of all bounded linear operators acting on a Banach space X. For T ∈ L(X ), let T ∗ ,N (T ), R(T ),σ(T ) and σ a (T ) denote the adjoint, the null space, the range, the spectrum and the approximate point spectrum of T respectively. Let α(T ) and β(T ) be the nullity and the deficiency of T defined by α(T ) = dimN (T ) and β(T ) = codimR(T ). If the range R(T ) of T is closed and α(T ) < ∞ (resp. β(T ) < ∞), then T is called an upper (resp. a lower) semi-Fredholm operator. In the sequel SF + (X ) will denote the set of all upper semi-Fredholm operators. If T ∈ L(X ) is either upper or a lower semi-Fredholm operator, then T is called a semi-Fredholm operator, and the index of T is defined by ind(T )= α(T ) - β(T ). If both α(T ) and β(T ) are finite, then T is a called a Fredholm operator. An operator T is called a Weyl operator if it is a Fredholm operator of index zero. The Weyl spectrum of T is defined by σ w (T )= {λ ∈ C : T - λI is not Weyl }. For T ∈ L(X ), let * Corresponding author. The second author was supported by Protars D11/16 and PGR- UMP.