Maps preserving the ε-Pseudo Spectrum of some product of operators

2023/05/04 by Hamze Bagherinejad, Bagherinejad, Hamze, Ali Iloon Kashkooly +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings, Modules, and Algebras #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2305.02683

openalex publication_date 2023/05/04 · openalex created_date 2023/05/07 · openalex updated_date 2026/07/28

Abstract

Let B(H) be the algebra of all bounded linear operators on infinite-dimensional complex Hilbert space H. For T, S ∈ B(H) denote by T\bullet S=TS+ST^∗ and [T∘ S]_∗=TS-ST^∗ the Jordan ∗-product and the skew Lie product of T and S, respectively. Fix ε > 0 and T ∈ B(H), let σ_ε(T) denote the ε-pseudo spectrum of T. In this paper, we describe bijective maps φ on B(H) which satisfy σ_ε([T₁\bullet T₂,T₃]_∗)=σ_ε([φ(T₁)\bullet φ(T₂),φ(T₃)]_∗), for all T₁, T₂, T₃ ∈ B(H). We also characterize bijective maps φ: B(H) → B(H) that satisfy σ_ε(T₁\diamond T₂∘_∗ T₃)=σ_ε(φ(T₁)\diamond φ(T₂)∘_∗ φ(T₃)), for all T₁, T₂, T₃ ∈ B(H), where T₁\diamond T₂=T₁T₂^∗+T₂^∗T₁ and T₁∘_∗ T₂=T₁T₂^∗-T₂T₁.

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