Analytic model of doubly commuting contractions

Abstract

An n-tuple (n ≥ 2), T = (T1, …,Tn), of commuting bounded linear operators on a Hilbert space H is doubly commuting if (formula presented) for all 1 ≤ i < j ≤ n. If in addition, each Ti ∈ C0, then we say that T is a doubly commuting pure tuple. In this paper we prove that a doubly commuting pure tuple T can be dilated to a tuple of shift operators on some suitable vector-valued Hardy space (formula presented) (Dn). As a consequence of the dilation theorem, we prove that there exists a closed subspace IT of the form (formula presented) such that H ≅ IT⊥ and (formula presented) where (formula presented) are Hilbert spaces and each (formula presented), 1 ≤ i ≤ n is either a one variable either a one variable inner function in zi, or the zero function.