Analytic projections, Corona problem and geometry of holomorphic vector bundles

Treil, Sergei; Wick, Brett

doi:10.1090/S0894-0347-08-00611-5

Analytic projections, Corona problem and geometry of holomorphic vector bundles
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by Sergei Treil and Brett D. Wick

J. Amer. Math. Soc. 22 (2009), 55-76

DOI: https://doi.org/10.1090/S0894-0347-08-00611-5

Published electronically: July 31, 2008

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Abstract:

The main result of the paper is a theorem giving a sufficient condition for the existence of a bounded analytic projection onto a holomorphic family of generally infinite dimensional subspaces (a holomorphic sub-bundle of a trivial bundle). This sufficient condition is also necessary in the case of finite dimension or codimension of the bundle. A simple lemma of N. Nikolski connects the existence of a bounded analytic projection with the Operator Corona Problem (existence of a bounded analytic left inverse for an operator-valued function), so as corollaries of the main result we obtain new results about the Operator Corona Problem. In particular, we find a new sufficient condition, a complete solution in the case of finite codimension, and a solution of the generalized Corona Problem.

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