Operators affiliated to the free shift on the free Hardy space

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").