Hildebrandt'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 \(A\).
Keywords: essential spectrum, essential numerical range, Hildebrandt's theorem.
Mathematics Subject Classification: 47A10, 47A12.