A formula of A-spectral radius for A^(1)/(2)-adjoint operators on semi-Hilbertian spaces

2024/02/20 by Majumdar, Arup, Johnson, P. Sam
#46C05 #47A10 #47A30 #47A80 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2402.12961

Abstract

In this paper, we prove the relation \fracr_A(T) + r_A(T^\diamond) + |r_A(T^\diamond) - r_A(T)|2 = sup \ |λ|: λ∈ σ_A(T)\, where A is a positive semidefinite operator (not necessarily to have a closed range) and r_A(T) is the A-spectral radius of T in B_A^(1)/(2)(H). Also we prove that sup \ |λ|: λ∈ σ_A(T)\ = r_A(T), when T ∈ B_A^(1)/(2)(H) \text commutes with A. By introducing A-Harte spectrum σ_A_h(T) of a d-tuple operator T= (T₁,…,T_d) ∈ (B_A^(1)/(2)(H))^d, we prove that r_A_h(T) ≤ sup \‖λ‖₂: λ∈ σ_A_h(T)\, where r_A_h(T) is the A-Harte spectral radius of T.

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