Joint similarity and dilations for noncontractive sequences of operators

Abstract

A characteristic function Θ T is defined, in terms of multianalytic operators on Fock spaces, for any noncontractive sequence T≔(T 1,…,T d) ( d∈ N or d=∞) of operators on a Hilbert space H . It is shown that if Θ T is bounded, then it is unitarily equivalent to a compression of an orthogonal projection (on Kreı̆n spaces). This leads to a generalization of a theorem of Davis and Foiaş, to multivariable setting. More precisely, it is proved that if T has bounded characteristic function, then it is jointly similar to a contractive sequence of operators, i.e., there exists a similarity S∈ B( H) such that the operator defined by the row matrix [ST 1S −1 ST 2S −1…ST dS −1] is a contraction. Motivated by the similarity problem, a multivariable dilation theory on Fock spaces with indefinite metric is developed for noncontractive d-tuples of operators. Wold-type decompositions for sequences of bounded isometries on Kreı̆n spaces and Fourier representations for d-orthogonal shifts are obtained and used to study the geometry of the canonical minimal isometric dilation associated with a sequence T of operators on a Hilbert space.