On the convexity of the quaternionic essential numerical range

TL;DR

This paper proves that the essential numerical range of a bounded operator on a quaternionic Hilbert space is convex, extending classical results and providing a quaternionic analogue of Lancaster's theorem.

Contribution

It establishes the convexity of the quaternionic essential numerical range and introduces a quaternionic version of Lancaster's theorem.

Findings

Essential numerical range is convex in quaternionic Hilbert spaces

Provides a quaternionic analogue of Lancaster's theorem

Extends classical numerical range results to quaternionic setting

Abstract

The numerical range in the quaternionic setting is, in general, a non convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.