In this note we study the property (w), a variant of Weyl's theorem introduced by Rakočević, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (w) holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property (w) for T (respectively T * ) coincide whenever T * (respectively T ) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property (w). Definitions and basic resultsThroughout this paper, X denotes an infinite-dimensional complex Banach space, L(X) the algebra of all bounded linear operators on X. For an operator T ∈ L(X) we shall denote by α(T ) the dimension of the kernel ker T , and by β(T ) the codimension of the range T (X). Let Φ + (X) := T ∈ L(X): α(T ) < ∞ and T (X) is closed

A bounded operator defined on a Banach space is said to be polaroid if every isolated point of the spectrum is a pole of the resolvent. In this paper we consider the two related notions of left and right polaroid, and explore them together with the condition of being a-polaroid. Moreover, the equivalences of Weyl type theorems and generalized Weyl type theorems are investigated for left and a-polaroid operators. As a consequence, we obtain a general framework which allows us to derive in a unified way many recent results, concerning Weyl type theorems (generalized or not) for important classes of operators.Mathematics Subject Classification (2010). Primary 47A10, 47A11. Secondary 47A53, 47A55.

Abstract. The left Drazin spectrum and the Drazin spectrum coincide with the upper semi-B-Browder spectrum and the B-Browder spectrum, respectively. We also prove that some spectra coincide whenever T or T * satisfies the single-valued extension property.

Preface page vii Part I Banach algebras H. Garth Dales 1 Definitions and examples 2 Ideals and the spectrum 3 Gelfand theory 4 The functional calculus 5 Automatic continuity of homomorphisms 6 Modules and derivatives 7 Cohomology Part II Harmonic analysis and amenability George A. Willis 8 Locally compact groups 9 Group algebras and representations 10 Convolution operators 11 Amenable groups 12 Harmonic analysis and automatic continuity Part III Invariant subspaces Jörg Eschmeier 13 Compact operators 14 Unitary dilations and the H ∞-functional class 15 Hyperinvariant subspaces 16 Invariant subspaces for contractions 17 Invariant subspaces for subnormal operators 18 Invariant subspaces for subdecomposable operators 19 Reflexivity of operator algebras 20 Invariant subspaces for commuting contractions Appendix to Part III Part IV Local spectral theory Kjeld Bagger Laursen 21 Basic notions from operator theory 22 Classes of decomposable operators v vi Contents 23 Duality theory 24 Preservation of spectra and index 25 Multipliers on commutative Banach algebras Appendix to Part IV Part V Single-valued extension property and Fredholm theory Pietro Aiena 26 Semi-regular operators 27 The single-valued extension property 28 SVEP for semi-Fredholm operators Index of symbols Subject index

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