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On tuples of commuting operators in positive semidefinite inner product spaces. (English) Zbl 1518.47012

Summary: This paper is concerned with the study of certain properties of operator tuples on a complex Hilbert space \(\mathcal{H}\) when a semi-inner product induced by a positive operator \(A\) on \(\mathcal{H}\) is considered. In particular, we show that \(r_A(\mathbf{T})\leq\omega_A(\mathbf{T})\) for every commuting operator tuple \(\mathbf{T}=(T_1,\dots,T_d)\) such that each \(T_k\) admits an \(A\)-adjoint operator, where \(r_A(\mathbf{T})\) and \(\omega_A(\mathbf{T})\) denote respectively the \(A\)-joint spectral radius and the \(A\)-joint numerical radius of T. This study allows to establish that \(r_A(\mathbf{T})=\omega_A(\mathbf{T})=\|\mathbf{T}\|_A\) for every \(A\)-normal commuting tuple of operators T, where \(\| \mathbf{T} \|_A\) is denoted to be the \(A\)-joint operator seminorm of T. In addition, the \(A\)-joint spectral radius of an \((A, m)\)-isometric tuple of commuting operators is studied.

MSC:

47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A12 Numerical range, numerical radius
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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