The spectrum of the product of operators, and the product of their numerical ranges

Abstract

We show that a compact operator A is a multiple of a positive semi-definite operator if and only if sigma(AB) subset of <(W(A)W(B))over bar>, for all (rank one) operators B. An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators A so that equivalence conditions hold. (C) 2014 Elsevier Inc. All rights reserved.