Free outer functions in complete Pick spaces

Aleman, Alexandru; Hartz, Michael; McCarthy, John; Richter, Stefan

doi:10.1090/tran/8812

Free outer functions in complete Pick spaces
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by Alexandru Aleman, Michael Hartz, John E. McCarthy and Stefan Richter HTML | PDF

Trans. Amer. Math. Soc. 376 (2023), 1929-1978 Request permission

Abstract:

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function $f$ in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type $f=\varphi g$, where $g$ is cyclic, $\varphi$ is a contractive multiplier, and $\|f\|=\|g\|$. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization.

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