Dilation Theory and Functional Models for Tetrablock Contractions

Abstract

A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator T can be dilated to a unitary U, i.e., T-n = PHUn|H for all n = 0, 1, 2, .... A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain omega contained in Cd, (ii) the contraction operator T is replaced by an omega-contraction, i.e., a commutative operator d-tuple T = (T-1, ... , T-d) on a Hilbert space H such that Ilr(T1, . . . , Td)IIL(H) <= sup(lambda is an element of omega) |r(lambda)| for all rational functions with no singularities in omega and the unitary operator U is replaced by an omega-unitary operator tuple, i.e., a commutative operator d-tuple U = (U-1, ... , U-d) of commuting normal operators with joint spectrum contained in the distinguished boundary b omega of omega. For a given domain omega subset of C-d, the rational dilation question asks: given an omega-contraction T on H, is it always possible to find an omega-unitary U on a larger Hilbert space K superset of H so that, for any d-variable rational function without singularities in omega, one can recover r(T) as r(T) = P(H)r(U)|(H). We focus here on the case where (sic)omega = E, a domain in C-3 called the tetrablock. (i) We identify a complete set of unitary invariants for a E-contraction (A, B, T) which can then be used to write down a functional model for (A, B, T), thereby extending earlier results only done for a special case, (ii) we identify the class of pseudo-commutative E-isometries (a priori slightly larger than the class of E-isometries) to which any E-contraction can be lifted, and (iii) we use our functional model to recover an earlier result on the existence and uniqueness of a E-isometric lift (V-1, V-2, V-3) of a special type for a E-contraction (A, B, T).