Hilderbrand's theorem for the essential spectrum

Abstract

We prove a variant of Hildebrandt’s theorem which asserts that the convex hull of the essential spectrum of an operator $A$ on a complex Hilbert space is equal to the intersection of the essential numerical ranges of operators which are similar to $A$. As a consequence, it is given a necessary and sufficient condition for zero not being in the convex hull of the essential spectrum of $A$.