# Multi-Toeplitz operators and free pluriharmonic functions

## TL;DR

This paper develops a noncommutative framework for multi-Toeplitz operators on Fock spaces, establishing analogues of classical harmonic function results and introducing a free analytic calculus for operator n-tuples.

## Contribution

It introduces weighted multi-Toeplitz operators associated with noncommutative domains and characterizes bounded free pluriharmonic functions as Berezin transforms of these operators.

## Key findings

- Bounded free pluriharmonic functions are Berezin transforms of weighted multi-Toeplitz operators.
- Solves the Dirichlet extension problem in the noncommutative setting.
- Provides a free analytic functional calculus extending to free pluriharmonic functions.

## Abstract

We initiate the study of weighted multi-Toeplitz operators associated with noncommutative regular domains in B(H)^n. These operators are acting on the full Fock space with n generators and have as symbols free pluriharmonic functions. Several classical results from complex analysis concerning harmonic functions have analogues in our noncommutative setting. In particular, we show that the bounded free pluriharmonic functions are precisely those which are noncommutative Berezin transforms of weighted multi-Toeplitz operators, and solve the Dirichlet extension problem in this setting. Using noncommutative Cauchy transforms, we provide a free analytic functional calculus for n-tuples of operators, which extends to free pluriharmonic functions.