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

We study the ideal membership problem in $H^{\mathrm{\infty }}$ on the unit disc. Thus, given functions $f,f_1,\dots ,f_n$ in $H^{\mathrm{\infty }}$, we seek sufficient conditions on the size of $f$ in order for $f$ to belong to the ideal of $H^{\mathrm{\infty }}$ 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 $H^2$.