Duality, Tangential Interpolation, and Toeplitz Corona Problems

TL;DR

This paper extends a method to prove a tangential interpolation theorem for subalgebras of H-infinity, leading to a Toeplitz corona theorem with matrix positivity conditions indexed by cyclic subspaces.

Contribution

It introduces a new approach to tangential interpolation for subalgebras of H-infinity and establishes a Toeplitz corona theorem with novel indexing of positivity conditions.

Findings

Matrix positivity conditions are indexed by cyclic subspaces.

The method generalizes previous results for the ball and polydisk algebra.

A new proof technique for Toeplitz corona problems is developed.

Abstract

In this paper we extend a method of Arveson and McCullough to prove a tangential interpolation theorem for subalgebras of $H^\infty$. This tangential interpolation result implies a Toelitz corona theorem. In particular, it is shown that the set of matrix positivity conditions is indexed by cyclic subspaces, which is analogous to the results obtained for the ball and the polydisk algebra by Trent-Wick and Douglas-Sarkar.