  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 $\mathbf{T}\in C_{1,N}$.
