Ideal membership in $H^{\infty }$: Toeplitz corona approach

Abstract

We study the ideal membership problem in $H^{\infty }$ on the unit disc. Thus, given functions $f,f_1,\dots ,f_n$ 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 $f_1,\dots ,f_n$. 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 $H^2$, which is equivalent to the ideal membership problem by the Nevanlinna-Pick property of $H^2$.

16 pages; minor changes

References in corpus (3)

• The Corona Theorem for the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in ${\mathbb{C}}^n$
• The Matrix-Valued $H^p$ Corona Problem in the Disk and Polydisk
• The problem of ideals of $H^{\infty }$: beyond the exponent 3/2