A note on the maximal numerical range

TL;DR

This paper characterizes when the maximal numerical range of an operator intersects with its numerical range boundary, showing it occurs precisely for normaloid operators, and describes this intersection.

Contribution

It provides a new characterization of the intersection between the maximal numerical range and the boundary of the numerical range for operators, specifically identifying normaloid operators.

Findings

Maximal numerical range intersects the boundary only for normaloid operators.

Provides a description of the intersection between these ranges.

Characterizes normaloid operators via this intersection property.

Abstract

We show that the maximal numerical range of an operator has a non-empty intersection with the boundary of its numerical range if and only if the operator is normaloid. A description of this intersection is also given.