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Ideal membership in $H^\infty$: Toeplitz corona approach (1709.07939v2)
Published 22 Sep 2017 in math.CV and math.FA
Abstract: We study the ideal membership problem in $H\infty$ on the unit disc. Thus, given functions $f,f_1,\ldots,f_n$ in $H\infty$, we seek sufficient conditions on the size of $f$ in order for $f$ to belong to the ideal of $H\infty$ generated by $f_1,\ldots,f_n$. We provide a different proof of a theorem of Treil, which gives the sharpest known sufficient condition. To this end, we solve a closely related problem in the Hilbert space $H2$, which is equivalent to the ideal membership problem by the Nevanlinna-Pick property of $H2$.