Linear preservers of parallel matrix pairs with respect to the $k$-numerical radius

TL;DR

This paper classifies linear transformations on matrices that preserve specific pairs related to the $k$-numerical radius, revealing they are mostly scalar multiples of $w_k$-isometries with some exceptions.

Contribution

It provides a complete classification of linear preservers of parallel and TEA pairs with respect to the $k$-numerical radius on matrix spaces.

Findings

Preservers are scalar multiples of $w_k$-isometries.

Exceptional maps exist on $ ext{H}_n$ when $n=2k$.

Classification applies to both $ ext{M}_n$ and $ ext{H}_n$.

Abstract

Let $1 \leq k < n$ be integers. Two $n \times n$ matrices $A$ and $B$ form a parallel pair with respect to the $k$-numerical radius $w_k$ if $w_k(A + \mu B) = w_k(A) + w_k(B)$ for some scalar $\mu$ with $|\mu| = 1$; they form a TEA (triangle equality attaining) pair if the preceding equation holds for $\mu = 1$. We classify linear bijections on $\mathbb M_n$ and on $\mathbb H_n$ which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of $w_k$-isometries, except for some exceptional maps on $\mathbb H_n$ when $n=2k$.