Dilation, functional model and a complete unitary invariant for C.0Γn-contractions

Abstract

A commuting tuple of operators [Formula: see text], defined on a Hilbert space [Formula: see text], for which the closed symmetrized polydisc [Formula: see text] is a spectral set, is called a [Formula: see text]-contraction. A [Formula: see text]-contraction is said to be pure or [Formula: see text] if [Formula: see text] is [Formula: see text], that is, if [Formula: see text] strongly as [Formula: see text]. We show that for any [Formula: see text]-contraction [Formula: see text], there is a unique operator tuple [Formula: see text] that satisfies the operator identities [Formula: see text] This unique tuple is called the fundamental operator tuple or [Formula: see text]-tuple of [Formula: see text]. With the help of the [Formula: see text]-tuple, we construct an operator model for a [Formula: see text] [Formula: see text]-contraction and show that there exist [Formula: see text] operators [Formula: see text] such that each [Formula: see text] can be represented as [Formula: see text]. We find an explicit minimal dilation for a class of [Formula: see text] [Formula: see text]-contractions whose [Formula: see text]-tuples satisfy a certain condition. Also, we establish that the [Formula: see text]-tuple of [Formula: see text] together with the characteristic function of [Formula: see text] constitutes a complete unitary invariant for the [Formula: see text] [Formula: see text]-contractions. The entire program is an analog of the Sz.-Nagy–Foias theory for [Formula: see text] contractions.