Von Neumann Inequality and Dilation Theory on Regular $${{\mathcal {U}}}$$ -Twisted Polyballs

Abstract

The goal of this paper is to develop a dilation theory on the regular \({{\mathcal {U}}}\)-twisted polyball \(\textbf{B}_{{\mathcal {U}}}({{\mathcal {H}}})\), which is the set of all k-tuples \(T:=(T_1,\ldots , T_k)\) of row contractions \(T_i:=[T_{i,1}\cdots T_{i,n_i}]\) on a Hilbert space \({{\mathcal {H}}}\) satisfying certain positivity condition on the defect operator \(\Delta _T(I)\) and \({{\mathcal {U}}}\)-commutation relations, where \({{\mathcal {U}}}\subset B({{\mathcal {H}}})\) is an appropriate set of unitary operators. The role of operator model is played by a standard multi-shift \(\textbf{S}:=(\textbf{S}_1,\ldots , \textbf{S}_k)\) with \(\textbf{S}_i:=[\textbf{S}_{i,1}\cdots \textbf{S}_{i,n_i}]\), which is a k-tuple of doubly \(I\otimes {{\mathcal {U}}}\)-commuting pure row isometries on the Hilbert space \(\ell ^2({{\mathbb {F}}}_{n_1}^+\times \cdots \times {{\mathbb {F}}}_{n_k}^+)\otimes {{\mathcal {H}}}\), where \({{\mathbb {F}}}_{n_i}^+\) is the unital free semigroup with \(n_i\) generators and each \(\textbf{S}_{i,j}\) is a weighted shift with operator-valued weights. It is shown that many of the classical results concerning the dilation theory of contractions on Hilbert spaces have analogues for \({{\mathcal {U}}}\)-twisted polyballs. This includes: Sz.-Nagy dilation theorem, von Neumann inequality, Itô and Brehmer dilations for commuting isometries and contractions, respectively, and Beurling characterization of the invariant subspaces for the unilateral shift on the Hardy space \(H^2\).