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For a Hilbert function space $\mathcal H$ the Smirnov class $\mathcalN^+(\mathcal H)$ is defined to be the set of functions expressible as a ratioof bounded multipliers of $\mathcal H$, whose denominator is cyclic for theaction of $Mult(\mathcal H)$. It is known that for spaces $\mathcal H$ withcomplete Nevanlinna-Pick (CNP) kernel, the inclusion $\mathcal H\subset\mathcal N^+(\mathcal H)$ holds. We give a new proof of this fact, whichincludes the new conclusion that every $h\in\mathcal H$ can be expressed as aratio $b/a\in\mathcal N^+(\mathcal H)$ with $1/a$ already belonging to$\mathcal H$.  The proof for CNP kernels is based on another Smirnov-type result ofindependent 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 Smirnovclass} $\mathcal N^+_{left}$ and show that every $H \in \mathfrak F^2_d$belongs to it. The proof of the Smirnov theorem for CNP kernels is thenobtained by lifting holomorphic functions on the ball to free holomorphicfunctions, and applying the free Smirnov theorem.

The Smirnov classes for the Fock space and complete Pick spaces