Abstract

The Smirnov class for the classical Hardy space is the set of ratios of bounded analytic functions in 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 Cd. Identifying the Fock space with the free multi-variable Hardy space of non-commutative or free holomorphic functions in a non-commutative open unit ball in several matrix-variables, we prove that any closed, densely-defined operator affiliated to the right free multiplier algebra of the full Fock space acts as right multiplication by a non-commutative function in the right free Smirnov class (and analogously, replacing ‘right’ with ‘left’).

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