Composition operators on noncommutative Hardy spaces

TL;DR

This paper explores the properties of composition operators on noncommutative Hardy spaces, extending classical results to a free, multivariable operator setting, and highlighting the interaction between noncommutative and classical function theories.

Contribution

It introduces the study of composition operators on noncommutative Hardy spaces and establishes analogues of classical results in this new multivariable noncommutative context.

Findings

Boundedness and norm estimates for composition operators

Spectral properties and compactness results

Connections between noncommutative and classical function theories

Abstract

In this paper we initiate the study of composition operators on the noncommutative Hardy space $H^2_{\bf ball}$. Several classical results about composition operators (boundedness, norm estimates, spectral properties, compactness, similarity) have free analogues in our noncommutative multivariable setting. The most prominent feature of this paper is the interaction between the noncommutative analytic function theory in the unit ball of $B(\cH)^n$, the operator algebras generated by the left creation operators on the full Fock space with $n$ generators, and the classical complex function theory in the unit ball of $\CC^n$.