A joint similarity problem for \(n\)-tuples of operators on vector-valued Bergman spaces
Let \(A_1,\dots,A_n\) be bounded linear operators on a separable Hilbert space \(\mathcal H\). The \(n\)-tuple \((A_1,\dots,A_n)\) is said to be jointly similar to an \(n\)-tuple of contractions if there there exist an invertible operator \(V\in{\mathcal B}({\mathcal H})\), and \(S_i\in{\mathcal B}({\mathcal H})\) with \(\| S_i\|\leqslant 1\), such that \(A_i=V^{-1}S_iV\), \(1\leqslant i\leqslant n\). Let \(\mathbb D^d\) be the polydisc in \(\mathbb C^d\). A commuting \(d\)-tuple \((A_1,\dots,A_d)\) of bounded linear operators on \(\mathcal H\) is said to be jointly polynomially bounded if there exists a positive constant \(k\) such that, for any analytic polynomial \(p\) of \(d\) variables, the following inequality holds: \[ \| p(A_1,\dots,A_d)\|\leqslant k\sup\{| p(z)|: z\in{\mathbb C}^d\}. \] A commuting \(d\)-tuple \((A_1,\dots,A_d)\) is said to be jointly completely polynomially bounded if there exists a positive constant \(k\) such that \[ \left|\sum_{1\leqslant i,j\leqslant n}\langle p_{ij}(A_1,\dots,A_d)x_i,y_j\rangle\right|\leqslant k\| P\|_{M_n(\mathcal A(\mathbb D^d))}\left( \sum_{i=1}^n\| x_i\|^2\right)^{1/2}\left( \sum_{j=1}^n\| y_j\|^2\right)^{1/2} \] for any \(n\times n\) matrix \(P=(p_{ij})\) of analytic polynomials of \(d\) variables and an arbitrary positive integer \(n\). Here, \(\| P\|_{M_n(\mathcal A(\mathbb D^d))}=\sup\{\| p_{ij}(z)\|: z\in{\mathbb D^d}\}\). The paper deals with operators of the form \[ R_T=\begin{pmatrix} M^\ast_z&\Gamma_T \\ 0&M_z\end{pmatrix}, \] where \(M_z\) denotes the operator of multiplication by \(z\) on the standard vector-valued Bergman space \(L_a^{2,\alpha}(\mathcal H)\), \(\alpha>-1\), and \(\Gamma_T\) is a Hankel operator with the analytic symbol \(T: {\mathbb D}\rightarrow{\mathcal B}({\mathcal H})\). Since the Hankel operator \(\Gamma_T\) satisfies \(\Gamma_TM_z=M^\ast_z\Gamma_T\), any two operators \(R_{T_i}\), \(i=1,2\), commute. The main result of the paper is the following Theorem. Let \(T_i:\mathbb D\rightarrow{\mathcal B} ({\mathcal H})\) be holomorphic operator-valued functions such that \(\sup\{(1-| z|^2)\| T'_i(z)\|<\infty\), \(1\leqslant i\leqslant d\). Then the following are equivalent: {\parindent=7mm \begin{itemize}\item[(1)] The operators \(R_{T_i}\), \(1\leqslant i\leqslant d\), are polynomially bounded. \item[(2)] The \(d\)-tuple \((R_{T_1},\dots,R_{T_d})\) is jointly polynomially bounded. \item[(3)] The \(d\)-tuple \((R_{T_1},\dots,R_{T_d})\) is jointly completely polynomially bounded. \item[(4)] The \(d\)-tuple \((R_{T_1},\dots,R_{T_d})\) is jointly similar to a \(d\)-tuple of contractions. \item[(5)] The \(d\)-tuple \((R_{T_1},\dots,R_{T_d})\) is jointly similar to a \(d\)-tuple of contractions having a \(d\)-tuple of commuting unitary dilations. \item[(6)] \(\sup\{(1-| z|^2)\| T''_i(z)\|<\infty\), \(1\leqslant i\leqslant d\). \end{itemize}}
joint similarity to an \(n\)-tuple of contractions
scientific article; zbMATH DE number 5688810