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RESEARCH PRODUCT

Classes of operators satisfying a-Weyl's theorem

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In this article Weyl's theorem and a-Weyl's theorem on Banach spaces are related to an important property which has a leading role in local spectral theory: the single-valued extension theory. We show that if T has SVEP then Weyl's theorem and a-Weyl's theorem for T are equivalent, and analogously, if T has SVEP then Weyl's theorem and a-Weyl's theorem for T are equivalent. From this result we deduce that a-Weyl's theorem holds for classes of operators for which the quasi-nilpotent part H0(I T ) is equal to ker (I T ) p for some p2N and every 2C, and for algebraically paranormal operators on Hilbert spaces. We also improve recent results established by Curto and Han, Han and Lee, and Oudghiri. 1. Notation and terminology. We begin with some standard nota- tions in Fredholm theory. Throughout this note by L(X) we denote the algebra of all bounded linear operators acting on an innite-dimension al complex Banach space X. For every T 2 L(X) we denote by (T ) and (T ) the dimension of the kernel kerT and the codimension of the range T (X), respectively. The class of upper semi-Fredholm operators is dened by +(X) :=fT 2 L(X) : (T ) <1 and T (X) is closedg; whilst the class of lower semi-Fredholm operators is dened by (X) :=fT 2 L(X) : (T ) <1g: An operator T 2 L(X) is said to be semi-Fredholm if T 2 +(X)( (X), whilst the class of Fredholm operators is (X) := +(X)\ (X). The index of a semi-Fredholm operator is dened by indT := (T ) (T ). For a linear operator T the ascent p := p(T ) is dened as the smallest nonnegative integer p such that kerT p = kerT p+1 . If such an integer does not exist we put p(T ) =1. Analogously, the descent q := q(T ) is dened as the smallest nonnegative integer q such that T q (X) = T q+1 (X), and if such an integer does not exist we put q(T ) =1. A classical result states that if 2000 Mathematics Subject Classic ation: Primary 47A10, 47A11; Secondary 47A53, 47A55.