PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 8, August 2008, Pages 2839–2848 S 0002-9939(08)09138-7 Article electronically published on March 28, 2008 ON DRAZIN INVERTIBILITY PIETRO AIENA, MARIA T. BIONDI, AND CARLOS CARPINTERO (Communicated by Joseph A. Ball) Abstract. The left Drazin spectrum and the Drazin spectrum coincide with the upper semi-B-Browder spectrum and the B-Browder spectrum, respec- tively. We also prove that some spectra coincide whenever T or T ∗ satisfies the single-valued extension property. 1. Introduction and preliminaries Throughout this note L(X) will denote the algebra of all bounded linear op- erators acting on an infinite-dimensional complex Banach space X. The operator T ∈ L(X) is said to be upper semi-Fredholm if its kernel ker T is finite-dimensional and the range T (X) is closed, while T ∈ L(X) is said to be lower semi-Fredholm if T (X) is finite-codimensional. If either T is upper or lower semi-Fredholm, then T is said to be a semi-Fredholm operator, while T is said to be a Fredholm operator if it is both upper and lower semi-Fredholm. If T ∈ L(X) is semi-Fredholm, the classical index of T is defined by ind (T ) := dim ker T − codim T (X). The concept of semi-Fredholm operators has been generalized by Berkani ([9], [13] and [11]) in the following way: for every T ∈ L(X) and a nonnegative integer n let us denote by T [n] the restriction of T to T n (X) viewed as a map from the space T n (X) into itself (we set T [0] = T ). T ∈ L(X) is said to be semi-B-Fredholm, (resp. B-Fredholm, upper semi-B-Fredholm, lower semi-B-Fredholm,) if for some integer n ≥ 0 the range T n (X) is closed and T [n] is a semi-Fredholm operator (resp. Fredholm, upper semi-Fredholm, lower semi-Fredholm). In thiscase T [m] is a semi- Fredholm operator for all m ≥ n ([13]). This enables one to define the index of a semi-B-Fredholm operator as ind T = ind T [n] . A bounded operator T ∈ L(X) is said to be a Weyl operator if T is a Fredholm operator having index 0. A bounded operator T ∈ L(X) is said to be B-Weyl if for some integer n ≥ 0 the range T n (X) is closed and T [n] is Weyl. The Weyl spectrum and the B-Weyl spectrum are defined, respectively, by σ w (T ) := {λ ∈ C : λI − T is not Weyl} and σ bw (T ) := {λ ∈ C : λI − T is not B-Weyl}. Recall that the ascent of an operator T ∈ L(X) is defined as the smallest non- negative integer p := p(T ) such that ker T p = ker T p+1 . If such an integer does not Received by the editors November 3, 2006, and, in revised form, February 26, 2007. 2000 Mathematics Subject Classification. Primary 47A10, 47A11; Secondary 47A53, 47A55. Key words and phrases. Localized SVEP, B-Browder operators, Drazin invertibility. c 2008 American Mathematical Society 2839 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use