Linear maps preserving parallel matrix pairs with respect to the Ky-Fan $k$-norm

Abstract

Two bounded linear operators $A$ and $B$ are parallel with respect to a norm $\Vert \cdot \Vert$ if $\Vert A+\mu B\Vert =\Vert A\Vert +\Vert B\Vert$ for some scalar $\mu$ with $|\mu |=1$. Characterization is obtained for bijective linear maps sending parallel bounded linear operators to parallel bounded linear operators with respect to the Ky-Fan $k$-norms.

20 pages