Classification of doubly 𝒰-commuting row isometries

We present the structure of the k-tuples of doubly U-commuting row isometries and the C^*-algebras they generate, where U is a set of commuting unitary operators on a Hilbert space. We obtain Wold decompositions and use them to classify the k-tuples of doubly U-commuting row isometries up to a unitary equivalence. This study leads to the development of a dilation theory on the regular U-twisted polyball B_ U( H), which is the set of all k-tuples T:=(T_1,…, T_k) of row contractions T_i:=[T_i,1⋯ T_i,n_i] on a Hilbert space H satisfying certain positivity condition on the defect operator Δ_T(I) and U-commutation relations. It is shown that many of the classical results concerning the dilation theory of contractions on Hilbert spaces have analogues for U-twisted polyballs. This includes: Sz.-Nagy dilation theorem, von Neumann inequality, Ito 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.