Noncommutative weighted shifts, joint similarity, and function theory in several variables

Abstract

The goal of this paper is to study the structure of noncommutative weighted shifts, their properties, and to understand their role as models (up to similarity) for $n$-tuples of operators on Hilbert spaces as well as their implications to function theory on noncommutative (resp.commutative) Reinhardt domains. We obtain a Rota type similarity result concerning the joint similarity of $n$-tuples of operators to parts of noncommutative weighted multi-shifts and provide a noncommutative multivariable analogue of Foias-Pearcy model for quasinilpotent operators. The model noncommutative weighted multi-shift which is studied in this paper is the $n$-tuple $W=(W_1,\ldots, W_n)$, where $W_i$ are weighted left creation operators of the full Fock space with $n$ generators associated with a weight sequence $\boldsymbol μ=\{μ_β\}_{|β|\geq 1}$ of nonnegative numbers. We also represent the injective weighted multi-shifts $W_1,\ldots, W_n$ as ordinary multiplications by $Z_1,\ldots, Z_n$ on a Hilbert space of noncommutative formal power series. This leads naturally to analytic function theory in several complex variables. One of the goal for the remainder of the paper is to analyze the extent to which our noncommutative formal power series represent analytic functions in several noncommutative (resp. commutative) variables and to develop a functional calculus for arbitrary $n$-tuples of operators on a Hilbert space.