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

TL;DR

This paper characterizes bijective linear maps that preserve the property of operators being parallel with respect to the Ky-Fan $k$-norm, advancing understanding of structure-preserving transformations in operator theory.

Contribution

It provides a complete characterization of linear maps that preserve parallelism of operators under the Ky-Fan $k$-norm, a novel result in the study of norm-preserving maps.

Findings

Characterization of bijective linear maps preserving parallelism.

Extension of parallelism concept to Ky-Fan $k$-norms.

Insights into structure-preserving transformations in operator spaces.

Abstract

Two bounded linear operators $A$ and $B$ are parallel with respect to a norm $\|\cdot\|$ if $\|A+\mu B\| = \|A\| + \|B\|$ 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.