The Smirnov classes for the Fock space and complete Pick spaces

TL;DR

This paper explores the Smirnov classes in Hilbert function spaces, providing new proofs and extending results to free holomorphic functions, with implications for spaces with complete Nevanlinna-Pick kernels.

Contribution

It offers a new proof that functions in the Hilbert space can be expressed as ratios in the Smirnov class, and introduces the free Smirnov class for non-commutative holomorphic functions.

Findings

Every function in the Fock space belongs to the free Smirnov class.

New proof that functions in CNP kernel spaces can be expressed as ratios with specific properties.

Extension of Smirnov class concepts to free (non-commutative) holomorphic functions.

Abstract

For a Hilbert function space $\mathcal H$ the Smirnov class $\mathcal N^+(\mathcal H)$ is defined to be the set of functions expressible as a ratio of bounded multipliers of $\mathcal H$, whose denominator is cyclic for the action of $Mult(\mathcal H)$. It is known that for spaces $\mathcal H$ with complete Nevanlinna-Pick (CNP) kernel, the inclusion $\mathcal H\subset \mathcal N^+(\mathcal H)$ holds. We give a new proof of this fact, which includes the new conclusion that every $h\in\mathcal H$ can be expressed as a ratio $b/a\in\mathcal N^+(\mathcal H)$ with $1/a$ already belonging to $\mathcal H$. The proof for CNP kernels is based on another Smirnov-type result of independent interest. We consider the Fock space $\mathfrak F^2_d$ of free (non-commutative) holomorphic functions and its algebra of bounded (left) multipliers $\mathfrak F^\infty_d$. We introduce the (left) {\em free Smirnov class} $\mathcal N^+_{left}$ and show that every $H \in \mathfrak F^2_d$ belongs to it. The proof of the Smirnov theorem for CNP kernels is then obtained by lifting holomorphic functions on the ball to free holomorphic functions, and applying the free Smirnov theorem.