Operators affiliated to the free shift on the free Hardy space

TL;DR

This paper extends the concept of the Smirnov class to non-commutative free Hardy spaces, showing that operators affiliated with the free multiplier algebra act as multiplication by functions in the free Smirnov class.

Contribution

It generalizes the classical Smirnov class to the non-commutative setting of the full Fock space and characterizes affiliated operators as multipliers by functions in this class.

Findings

Operators affiliated to the free multiplier algebra are multiplication by free Smirnov class functions.

The result applies to both right and left free multipliers.

This extends classical function theory to non-commutative multivariable spaces.

Abstract

The Smirnov class for the classical Hardy space is the set of ratios of bounded analytic functions on the open complex unit disk with outer denominators. This definition extends naturally to the commutative and non-commutative multi-variable settings of the Drury-Arveson space and the full Fock space over $\mathbb C ^d$. Identifying the Fock space with the free multi-variable Hardy space of non-commutative or free holomorphic functions on the non-commutative open unit ball, we prove that any closed, densely-defined operator affiliated to the right free multiplier algebra of the full Fock space acts as right rmultiplication by a function in the right free Smirnov class (and analogously, replacing "right" with "left").