Notes on the numerical radius for adjointable operators on Hilbert C^*-modules

2025/02/27 by Jun Li, Ke Wu, Li, J. +3
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2502.20259

openalex publication_date 2025/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a Hilbert module H over a C^*-algebra, let L(H) be the set of all adjointable operators on H. For each T\inL(H), its numerical radius is defined by w(T)=sup\‖⟨ Tx, x ⟩‖: x∈ H, ‖x‖=1\. It is proved that w(T)=‖T‖ whenever T is normal. Examples are constructed to show that there exist Hilbert module H over certain C^*-algebra and T₁,T₂∈ L(H) with T₁²=0 such that w(T₁)≠ \frac12 ‖T₁‖ and sup_{θ∈ [0,2π]}‖Re(e^iθT₂)‖

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