Multivariable $ρ$-contractions

Abstract

We suggest a new version of the notion of $ρ$-dilation ($ρ>0$) 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 $C_{ρ,N}$ if $\mathbf{A}$ admits a $ρ$-dilation $\widetilde{\mathbf{A}}=(\widetilde{A}_1,...,\widetilde{A}_N)$ for which $ζ\widetilde{\mathbf{A}}:=ζ_1\widetilde{A}_1+... +ζ_N\widetilde{A}_N$ is a unitary operator for each $ζ:=(ζ_1,...,ζ_N)$ in the unit torus $\mathbb{T}^N$. For N=1 this class coincides with the class $C_ρ$ of B. Sz.-Nagy and C. Foiaş. We generalize the known descriptions of $C_{ρ,1}=C_ρ$ to the case of $C_{ρ,N}, N>1$, using so-called Agler kernels. Also, the notion of operator radii $w_ρ, ρ>0$, 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 $ρ>1$ and $N>1$ there exists an $\mathbf{A}=(A_1,...,A_N)\in C_{ρ,N}$ which is not simultaneously similar to any $\mathbf{T}=(T_1,...,T_N)\in C_{1,N}$, however if $\mathbf{A}\in C_{ρ,N}$ admits a uniform unitary $ρ$-dilation then $\mathbf{A}$ is simultaneously similar to some $\mathbf{T}\in C_{1,N}$.