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- W3134241570 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 $mathcal Hsubset mathcal N^+(mathcal H)$ holds. We give a new proof of this fact, which includes the new conclusion that every $hinmathcal H$ can be expressed as a ratio $b/ainmathcal N^+(mathcal H)$ 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." @default.
- W3134241570 created "2021-03-15" @default.
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- W3134241570 date "2021-01-01" @default.
- W3134241570 modified "2023-10-14" @default.
- W3134241570 title "The Smirnov classes for the Fock space and complete Pick spaces" @default.
- W3134241570 doi "https://doi.org/10.1512/iumj.2021.70.8203" @default.
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