Standard Operator Models in the Polydisc, II

We first show that every γ-contractive commuting multioperator is unitarily equivalent to the restriction of S⊕W to an invariant subspace, where S is a backwards multi-shift and W a γ-isometry. We then describe γ-isometries in terms of (γ, 1)-isometries, and establish that under an additional assumption on T , W above can be chosen to be a commuting multioperator of isometries. Our methods provide, as a by-product, a new proof of the existence of a regular unitary dilation for every (1, ..., 1)-contractive commuting multioperator.