IUMJ

Title: The Toeplitz corona problem for algebras of multipliers on a Nevanlinna-Pick space

Authors: Ryan Hamilton and Mrinal Raghupathi

Issue: Volume 61 (2012), Issue 4, 1393-1405

Abstract:

Suppose $\mathfrak{A}$ is an algebra of operators on a Hilbert space $H$ and $A_1,\dots,A_n\in\mathfrak{A}$. If the row operator $[A_1,\dots,A_n]$ has a right inverse, the Toeplitz corona problem asks if a right inverse can be found with entries in $\mathfrak{A}$. When $H$ is a complete Nevanlinna--Pick space and $\mathfrak{A}$ is a weakly closed algebra of multiplication operators on $H$, we show that under a stronger hypothesis, the Toeplitz corona problem for $\mathfrak{A}$ has a solution. When $\mathfrak{A}$ is the full multiplier algebra of $H$, the Toeplitz corona theorems of Arveson, Schubert, and Ball--Trent--Vinnikov are obtained. A tangential interpolation result for these algebras is introduced in order to solve the Toeplitz corona problem.