Joint Similarity to Operators in Noncommutative Varieties

In this paper, we solve several problems concerning joint similarity to n-tuples of operators in noncommutative varieties m ≥ 1, associated with positive regular free holomorphic functions f in n noncommuting variables and with sets of noncommutative polynomials in n indeterminates, where B(ℋ) is the algebra of all bounded linear operators on a Hilbert space ℋ. In particular, if f = X1 + … + Xn and = {0}, then the elements of the corresponding variety can be seen as noncommutative multivariable analogs of Agler’s m-hypercontractions. We introduce a class of generalized noncommutative Berezin transforms and use them to solve operator inequalities associated with noncommutative varieties . We point out a very strong connection between the cone of their positive solutions and the joint similarity problems. Several classical results concerning the similarity to contractions have analogs in our noncommutative multivariable setting. When consists of the commutators XiXj − XjXi, i, j∈{1, …, n}, we obtain commutative versions of these results. We remark that, in the particular case when n = m = 1, f = X, and = {0}, we recover the corresponding similarity results obtained by Sz.-Nagy, Rota, Foiaş, de Branges-Rovnyak, and Douglas. We use some of the results of this paper to provide Wold-type decompositions and triangulations for n-tuples of operators in noncommutative varieties , which parallel the classical Sz.-Nagy–Foiaş triangulations for contractions but also provide new proofs. As consequences, we prove the existence of joint invariant subspaces for certain classes of operators in .