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(Real)linear preservers of multiples of unitaries and matrix pairs with some extremal norm properties

Bojan Kuzma, Chi-Kwong Li, Edward Poon

TL;DR

This work classifies linear preservers of unitary multiples for square matrices over $\mathbb F$ and uses these structural results to characterize preservers of matrix pairs that achieve extremal norms under the spectral norm. For $n\ge 3$, bijective real-linear maps preserving unitary multiples must be of one of four canonical forms involving unitary factors and possible transposition/complex conjugation, with complex-linearity ruling out the latter two. The authors then obtain sharp descriptions of preservers of norm-multiplicative pairs, showing they must act by unitary conjugation (up to a scalar) or its complex-conjugate variant. The $2\times 2$ case, treated separately, exhibits a richer structure tied to a quaternionic subspace, yielding explicit parametrizations and showing that preservers can deviate from the higher-dimensional pattern. Overall, the results illuminate how unitary geometry constrains linear preservers and norm-extremal behavior in both large and small matrix dimensions.

Abstract

We determine the structure of linear maps on complex (real) square matrices sending unitary (orthogonal) matrices to multiples of unitary (orthogonal) matrices. The result is used to determine the linear preservers of matrix pairs satisfying the extremal norm properties $\|AB\| = \|A\| \|B\|$, $\|A^*B\| = \|A\| \|B\|$, or $\|AB^*\| = \|A\| \|B\|$, for the spectral norm $\|\cdot\|$.

(Real)linear preservers of multiples of unitaries and matrix pairs with some extremal norm properties

TL;DR

This work classifies linear preservers of unitary multiples for square matrices over and uses these structural results to characterize preservers of matrix pairs that achieve extremal norms under the spectral norm. For , bijective real-linear maps preserving unitary multiples must be of one of four canonical forms involving unitary factors and possible transposition/complex conjugation, with complex-linearity ruling out the latter two. The authors then obtain sharp descriptions of preservers of norm-multiplicative pairs, showing they must act by unitary conjugation (up to a scalar) or its complex-conjugate variant. The case, treated separately, exhibits a richer structure tied to a quaternionic subspace, yielding explicit parametrizations and showing that preservers can deviate from the higher-dimensional pattern. Overall, the results illuminate how unitary geometry constrains linear preservers and norm-extremal behavior in both large and small matrix dimensions.

Abstract

We determine the structure of linear maps on complex (real) square matrices sending unitary (orthogonal) matrices to multiples of unitary (orthogonal) matrices. The result is used to determine the linear preservers of matrix pairs satisfying the extremal norm properties , , or , for the spectral norm .

Paper Structure

This paper contains 9 sections, 16 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Real-linear preservers on matrices of size at least 3
  3. Real-linear preservers of multiples of unitary/orthogonal matrices
  4. Matrix pairs with product attaining maximum norm value
  5. Related results
  6. Real-linear preservers on $2$-by-$2$ matrices
  7. Preservers of multiples of unitary/orthogonal matrices
  8. Matrix pairs with product attaining maximum norm value
  9. Related results

Key Result

Theorem 2.1

Let $n\ge 3$ and let $T\colon M_n({\mathbb F})\to M_n({\mathbb F})$ be a real-linear map. The following conditions are equivalent. Moreover, if $T\colon M_n({\mathbb C})\to M_n({\mathbb C})$ is complex-linear then the last two forms do not exist.

Theorems & Definitions (33)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main1']}
  • Corollary 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 23 more