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

Abstract

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