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Maps on self-adjoint operators preserving some relations related to commutativity

Mahdi Karder, Tatjana Petek

Abstract

In this paper, we give the complete description of maps on self-adjoint bounded operators on Hilbert space which preserve a triadic relation involving the difference of operators and either commutativity or quasi-commutativity in both directions. We show that those maps are implemented by unitary or antiunitary equivalence and possible additive perturbation by a scalar operator.

Maps on self-adjoint operators preserving some relations related to commutativity

Abstract

In this paper, we give the complete description of maps on self-adjoint bounded operators on Hilbert space which preserve a triadic relation involving the difference of operators and either commutativity or quasi-commutativity in both directions. We show that those maps are implemented by unitary or antiunitary equivalence and possible additive perturbation by a scalar operator.
Paper Structure (3 sections, 14 theorems, 28 equations)

This paper contains 3 sections, 14 theorems, 28 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Proof of Theorem \ref{['th:4']}
  3. Proof of Theorem $\ref{['th:5']}$

Key Result

Theorem 1.1

molnar1 Let $\mathcal{H}$ be a complex separable Hilbert space with $\dim \mathcal{H} \geq 3$ and let $\phi:B_s(\mathcal{H} ) \mapsto B_s(\mathcal{H} )$ be a bijective transformation which preserves commutativity in both directions. Then there exists either a unitary or an antiunitary operator $U$ o where $\sigma(A)$ denotes the spectrum of $A$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 1.7
  • proof
  • Lemma 1.8
  • proof
  • ...and 12 more