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

### Abstract

Suppose $\fA$ is an algebra of operators on a Hilbert space $H$ and $A_1,..., A_n \in \fA$. If the row operator $[A_1,..., A_n] \in B(H^{(n)},H)$ has a right inverse in $B(H, H^{(n)})$, the Toeplitz corona problem for $\fA$ asks if a right inverse can be found with entries in $\fA$. When $H$ is a complete Nevanlinna-Pick space and $\fA$ is a weakly-closed algebra of multiplication operators on $H$, we show that under a stronger hypothesis, the corona problem for $\fA$ has a solution. When $\fA$ 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 developed in order to solve the Toeplitz corona problem.Comment: 13 page

Topics: Mathematics - Functional Analysis, Mathematics - Operator Algebras
Year: 2011
OAI identifier: oai:arXiv.org:1104.3821
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