A note on doubly commuting tuples of hyponormal operators on Hilbert spaces (Q2189572)

Let \(\mathcal{B(H)}\) be the \(C^{\ast}\)-algebra of bounded linear operators defined on a separable complex Hilbert space \(\mathcal{H}\). A~\(d\)-tuple \(T=(T_{1},\dots,T_{d}) \in \mathcal{B(H)}^{d}\) is said to be a commuting \(d\)-tuple if \(T_{i}T_{j}=T_{j}T_{i}\) for all \(i,j \in \{1,\dots,d\}\). It is called doubly commuting \(d\)-tuple if \(T_{i}T_{j}=T_{j}T_{i}\) and \(T_{i}^{\ast}T_{j}=T_{j}T_{i}^{\ast}\) for all \(1 \leq i\neq j \leq d\). Given a \(d\)-tuple \(T=(T_{1},\dots,T_{d}) \in \mathcal{B(H)}^{d}\), the joint operator norm of \(T\) is defined as \[ \|T\|:=\sup\Big{\{}\Big{(}\sum_{k=1}^{d}\|T_{k}x\|^{2}\Big{)}^{\frac{1}{2}}: x\in\mathcal{H}, \|x\|=1 \Big{\}}=\Big{\|}\sum_{k=1}^{d}T_{k}^{\ast}T_{k}\Big{\|}^{\frac{1}{2}}. \] In [Unitary invariants in multivariable operator theory. Providence, RI: American Mathematical Society (AMS) (2009; Zbl 1180.47010)], \textit{G. Popescu} introduced the following norm and joint spectral radius of a \(d\)-tuple \(T=(T_{1},\dots,T_{d}) \in \mathcal{B(H)}^{d}\) as \[ \|T\|_{e}:=\sup_{(\lambda_{1},\dots,\lambda_{d})\in \mathbb{B}_{d}}\|\lambda_{1}T_{1}+\cdots +\lambda_{d}T_{d}\| \] where \(\mathbb{B}_{d}\) denotes the open unit ball of \(\mathbb{C}^{d}\) with respect to the euclidean norm, and \[ r_{e}(T)=\sup_{(\lambda_{1},\dots,\lambda_{d})\in \mathbb{B}_{d}}r(\lambda_{1}T_{1}+\cdots +\lambda_{d}T_{d}) \] where \(r(\cdot)\) denotes the classical spectral radius, respectively. In [J. Funct. Anal. 57, 21--30 (1984; Zbl 0558.47004)], \textit{J. W. Bunce} introduced the algebraic joint spectral radius of a \(d\)-tuple \(T=(T_{1},\dots,T_{d}) \in \mathcal{B(H)}^{d}\) as \[ r(T)=\inf_{n \in \mathbb{N}^{\ast}}\Big{\|}\sum_{|\alpha|=n, \alpha \in \mathbb{N}^{d}}\frac{n!}{\alpha!}{T^{\ast}}^{\alpha}T^{\alpha}\Big{\|}^{\frac{1}{2n}}=\lim_{n\rightarrow}\Big{\|}\sum_{|\alpha|=n, \alpha \in \mathbb{N}^{d}}\frac{n!}{\alpha!}{T^{\ast}}^{\alpha}T^{\alpha}\Big{\|}^{\frac{1}{2n}} \] where for \(\alpha=(\alpha_{1},\dots,\alpha_{d}) \in \mathbb{N}^{d}\), \(\alpha!:=\prod_{k=1}^{d}\alpha_{k}!\), \(|\alpha|:=\sum_{j=1}^{d}|\alpha_{j}|\), \(T^{\ast}=(T_{1}^{\ast}, \dots,T_{d}^{\ast})\), and \(T^{\alpha}:=\prod_{k=1}^{d}T_{k}^{\alpha_{k}}\). In the present paper, it turns out that, if \(T=(T_{1},\dots,T_{d}) \in \mathcal{B(H)}^{d}\) is a commuting \(d\)-tuple of hyponormal operators, then \(\|T\|_{e}=\|T\|\) and \(r(T)=\|T\|=\omega(T)\) hold where \(\omega(T)\) denotes the joint numerical radius of \(T\). Moreover, the author proves that, if \(T=(T_{1},\dots,T_{d})\) and \(S=(S_{1},\dots,S_{d})\) in \(\mathcal{B(H)}^{d}\) are two commuting \(d\)-tuples of hyponormal operators on \(\mathcal{H}\), then \[ \|T-S\| \leq \sqrt{2}\max\{\|\lambda-\mu\|_{2}: \lambda \in \sigma_{H}(T), \mu\in\sigma_{H}(S) \} \] where \(\sigma_{H}(\cdot)\) denotes the Harte spectrum, and the above inequality is sharp.