Multivariable $\rho$-contractions

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 $\widetilde{\mathbf{A}}=(\widetilde{A}_1,...,\widetilde{A}_N)$ for which $\zeta\widetilde{\mathbf{A}}:=\zeta_1\widetilde{A}_1+... +\zeta_N\widetilde{A}_N$ 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. Foia\c{s}. 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 $\mathbf{T}\in C_{1,N}$.