A joint similarity problem on vector-valued Bergman spaces
Given a separable Hilbert space \({\mathcal H}\), let \({\mathcal B}({\mathcal H})\) denote the algebra of all bounded linear operators on \({\mathcal H}\). Let \({\mathbb D}\) be the open unit disk. A commuting pair of operators \((T_1,T_2)\) with \(T_j\in{\mathcal B}({\mathcal H})\) is said to be jointly polynomially bounded if there exists a positive constant \(k\) such that \(\|p(T_1,T_2)\|\leq k\sup_{z\in{\mathbb D}^2}\|p(z)\|\) for any analytic polynomial of two variables \(p\), and it is said to be jointly completely polynomially bounded if there exists a constant positive \(k\) such that \[ \Bigg|\sum_{1\leq i,j\leq n} \langle p_{ij}(T_1,T_2)x_i,y_j\rangle\Bigg| \leq k\sup_{z\in{\mathbb D}^2} \|(p_{ij}(z_1,z_2))\| \Bigg(\sum_{i=1}^n\|x_i\|^2\Bigg)^{1/2} \Bigg(\sum_{j=1}^n\|y_j\|^2\Bigg)^{1/2}, \] whenever \((p_{ij})_{1\leq i,j\leq n}\) is an \(n\times n\) matrix of analytic polynomials of two variables, \(n=1,2,\dots\), and \(\{x_i\}_{i=1}^n\), \(\{y_j\}_{j=1}^n\) are vectors in \({\mathcal H}\). Analogously, one can define the notions of a polynomially bounded operator and a completely polynomially bounded operator by substituting the pair of operators by one operator and the polynomials of two variables by polynomials of one variable in the above definitions. Consider the pair of operators \(R_T=\left(\begin{smallmatrix} M_z^* & \Gamma_T \\ 0 & M_z\end{smallmatrix}\right)\) and \(R_0=\left(\begin{smallmatrix} M_z^* & 0 \\ 0 & M_z\end{smallmatrix}\right)\) acting on the direct sum of weighted vector-valued Bergman spaces \(L_a^{2,\alpha}({\mathcal H})\oplus L_a^{2,\alpha}({\mathcal H})\), where \(M_z\) is the operator of multiplication by \(z\) on \(L_a^{2,\alpha}({\mathcal H})\) and \(\Gamma_T\) is a little Hankel operator. The main result of the paper is the following Theorem. Let \(T:{\mathbb D}\to{\mathcal B}({\mathcal H})\) be a holomorphic operator-valued function with \(\sup_{z\in{\mathbb D}}(1-|z|^2)\|T'(z)\|<\infty\). Then the following statements are equivalent: (i) the operator \(R_T\) is polynomially bounded; (ii) the pair \((R_T,R_0)\) is jointly polynomially bounded; (iii) the pair \((R_T,R_0)\) is jointly completely polynomially bounded; (iv) \(\sup_{z\in{\mathbb D}}(1-|z|^2)\|T''(z)\|<\infty\).
joint similarity to a pair of contractions
scientific article; zbMATH DE number 5561654