The Drazin spectrum of tensor product of Banach algebra elements and elementary operators

Abstract

Given unital Banach algebras A and B and elements a ∈ A and b ∈ B, the Drazin spectrum of will be fully characterized, where is a Banach algebra that is the completion of A ⊗ B with respect to a uniform crossnorm. To this end, however, first the isolated points of the spectrum of need to be characterized. On the other hand, given Banach spaces X and Y and Banach space operators S ∈ L(X) and T ∈ L(Y), using similar arguments the Drazin spectrum of τ ST ∈ L(L(Y, X)), the elementary operator defined by S and T, will be fully characterized.