Inverse continuity of the numerical range map for Hilbert space operators

TL;DR

This paper investigates the continuity properties of the inverse numerical range map for operators on Hilbert spaces, establishing strong continuity inside the numerical range and exploring boundary cases and special operator classes.

Contribution

It provides new results on the inverse map's continuity, especially inside the numerical range, and analyzes boundary behavior for different classes of operators.

Findings

Inverse map is strongly continuous inside the numerical range.

Strong and weak continuity can fail on the boundary.

Special cases like normal and compact operators are addressed.

Abstract

We describe continuity properties of the multivalued inverse of the numerical range map $f_A:x \mapsto \left\langle Ax, x \right\rangle$ associated with a linear operator $A$ defined on a complex Hilbert space $\mathcal{H}$. We prove in particular that $f_A^{-1}$ is strongly continuous at all points of the interior of the numerical range $W(A)$. We give examples where strong and weak continuity fail on the boundary and address special cases such as normal and compact operators.