Kato type operators and Weyl’s theorem

A Banach space operator T satisfies Weyl’s theorem if and only if T or T ∗ has SVEP at all complex numbers λ in the complement of the Weyl spectrum of T and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity. If T ∗ (respectively, T ) has SVEP and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity (respectively, T is Kato type at all λ ∈ isoσ(T )), then T satisfies a-Weyl’s theorem (respectively, T ∗ satisfies a-Weyl’s theorem).  2005 Published by Elsevier Inc