Functional calculus and multi-analytic models on regular Λ-polyballs

Abstract

In a recent paper, we introduced the standard k-tuple S:=(S1,…,Sk) of pure row isometries Si:=[Si,1⋯Si,ni] acting on the Hilbert space ℓ2(Fn1+×⋯×Fnk+), where Fn+ is the unital free semigroup with n generators, and showed that S is the universal k-tuple of doubly Λ-commuting row isometries, i.e.Si,s⁎Sj,t=λij(s,t)‾Sj,tSi,s⁎ for every i,j∈{1,…,k} with i≠j and every s∈{1,…,ni}, t∈{1,…,nj}, where Λij:=[λi,j(s,t)] is an ni×nj-matrix with the entries in T:={z∈C:|z|=1} and Λj,i=Λi,j⁎. It was also proved that the set of all k-tuples T:=(T1,…,Tk) of row operators Ti:=[Ti,1⋯Ti,ni] acting on a Hilbert space H which admit S as universal model, i.e. there is a Hilbert space D such that H is jointly co-invariant for all operators Si,s⊗ID andTi,s⁎=(Si,s⁎⊗ID)|H,i∈{1,…,k} and s∈{1,…,ni}, consists of the pure elements of a set BΛ(H) which was called the regular Λ-polyball. The goal of the present paper is to introduce and study noncommutative Hardy spaces associated with the regular Λ-polyball, to develop a functional calculus on noncommutative Hardy spaces for the completely non-coisometric (c.n.c.) k-tuples in BΛ(H), and to study the characteristic functions and the associated multi-analytic models for the c.n.c. elements in the regular Λ-polyball. In addition, we show that the characteristic function is a complete unitary invariant for the class of c.n.c. k-tuples in BΛ(H). These results extend the corresponding classical results of Sz.-Nagy–Foiaş for contractions and the noncommutative versions for row contractions. In the particular case when n1=⋯=nk=1 and Λij=1, we obtain a functional calculus and operator model theory in terms of characteristic functions for k-tuples of contractions satisfying Brehmer condition.