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From geometry to invertibility preservers

Hans Havlicek, Peter Šemrl

Abstract

We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.

From geometry to invertibility preservers

Abstract

We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.

Paper Structure

This paper contains 3 sections, 7 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. The finite-dimensional case
  3. The infinite-dimensional case

Key Result

Theorem 1.1

Let ${\mathbb F}$ be a field with at least three elements and $m,n$ integers with $m\ge n \ge 2$. Assume that $\phi : M_{m,n} \to M_{m,n}$ is a bijective map such that for every pair $A,B \in M_{m,n}$ we have $A\mathbin{\triangle} B$ if and only if $\phi (A) \mathbin{\triangle} \phi (B)$. Then the for every $A\in M_{m,n}$. If $m=n$, then we have the additional possibility that where $T,S,R \in

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3