Continuous Selections of the Inverse Numerical Range Map

TL;DR

This paper investigates the conditions under which the inverse numerical range map of a complex matrix admits a continuous single-valued selection, revealing that such selections exist for many matrices and identifying exceptions near boundary points.

Contribution

It characterizes when the inverse numerical range map has a continuous selection, including near boundary points, for a broad class of matrices.

Findings

Continuous selection exists for many matrices.

Exceptions occur near finite boundary points.

The inverse map is continuous except near certain boundary points.

Abstract

For a complex $n$-by-$n$ matrix $A$, the numerical range $F(A)$ is the range of the map $f_A(x) = x^*A x$ acting on the unit sphere in $\C^n$. We ask whether the multivalued inverse numerical range map $f_A^{-1}$ has a continuous single-valued selection defined on all or part of $F(A)$. We show that for a large class of matrices, $f_A^{-1}$ does have a continuous selection on $F(A)$. For other matrices, $f_A^{-1}$ has a continuous selection defined everywhere on $F(A)$ except in the vicinity of a finite number of exceptional points on the boundary of $F(A)$.