Multivariable $\rho$-contractions

Abstract

We suggest a new version of the notion of $\rho$-dilation ($\rho >0$) of an $N$-tuple $\mathbf{A}=(A_1,...,A_N)$ of bounded linear operators on a common Hilbert space. We say that $\mathbf{A}$ belongs to the class $C_{\rho ,N}$ if $\mathbf{A}$ admits a $\rho$-dilation $\tilde{\mathbf{A}}=({\tilde{A}}_1,...,{\tilde{A}}_N)$ for which $\zeta \tilde{\mathbf{A}}:={\zeta }_1{\tilde{A}}_1+...+{\zeta }_N{\tilde{A}}_N$ is a unitary operator for each $\zeta :=({\zeta }_1,...,{\zeta }_N)$ in the unit torus ${\mathbb{T}}^N$. For N=1 this class coincides with the class $C_{\rho }$ of B. Sz.-Nagy and C. Foiaş. We generalize the known descriptions of $C_{\rho ,1}=C_{\rho }$ to the case of $C_{\rho ,N},N>1$, using so-called Agler kernels. Also, the notion of operator radii $w_{\rho },\rho >0$, is generalized to the case of $N$-tuples of operators, and to the case of bounded (in a certain strong sense) holomorphic operator-valued functions in the open unit polydisk ${\mathbb{D}}^N$, with preservation of all the most important their properties. Finally, we show that for each $\rho >1$ and $N>1$ there exists an $\mathbf{A}=(A_1,...,A_N)\in C_{\rho ,N}$ which is not simultaneously similar to any $\mathbf{T}=(T_1,...,T_N)\in C_{1,N}$, however if $\mathbf{A}\in C_{\rho ,N}$ admits a uniform unitary $\rho$-dilation then $\mathbf{A}$ is simultaneously similar to some $\mathbf{T}\in C_{1,N}$.