# Journal of Operator Theory

Volume 78, Issue 1,  Summer  2017  pp. 3-20.

On $\rho$-dilations of commuting operators

Summary:  Let $n\geqslant 1$ and let $c_{F,G}$ be given real numbers defined for all pairs of disjoint subsets $F,G\subset\{1,\dots,n\}$. We characterize commuting $n$-tuples of operators $T=(T_1,\dots,T_n)$ acting on a Hilbert space $H$ which have a commuting unitary dilation $U=(U_1,\ldots,U_n)\in B(K)^n$, $K\supset H$ such that $P_HU^{*\beta}U^\alpha |_H= c_{\supp\alpha,\,\supp\beta} T^{*\beta}T^\alpha$ for all $\alpha,\beta\in\mathbb{Z}_+^n, \supp\,\alpha\cap\supp\,\beta=\emptyset$. This unifies and generalizes the concepts of $\rho$-dilations of a single operator and of regular unitary dilations of commuting $n$-tuples. We discuss also other interesting cases.
Keywords:  $\rho$-dilation, regular unitary dilation