Inverse continuity on the boundary of the numerical range

TL;DR

This paper studies the inverse of a quadratic form mapping on the unit sphere related to a matrix, analyzing its continuity properties and identifying conditions for failures and weak inverse continuity.

Contribution

It provides a detailed analysis of inverse continuity failures, offering necessary and sufficient conditions for specific matrix dimensions.

Findings

Finitely many inverse continuity failures can occur.

Conditions for inverse continuity failures are characterized.

Necessary and sufficient conditions for weak inverse continuity are established for n=4.

Abstract

Let $A \in M_n(\C)$. We consider the mapping $f_A(x)=x^*Ax$, defined on the unit sphere in $\C^n$. The map has a multi-valued inverse $f_A^{-1}$, and the continuity properties of $f_A^{-1}$ are considered in terms of the structure of the set of pre-images for points in the numerical range. It is shown that there may be only finitely many failures of continuity of $f_A^{-1}$, and conditions for where these failure occur are given. Additionally, we give a necessary and sufficient condition for weak inverse continuity to hold for $n=4$ and a sufficient condition for $n>4$.