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arXiv · math/0702756

Analytic projections, Corona Problem and geometry of holomorphic vector bundles

Sergei Treil·Brett Wick
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Abstract

The main result of the paper is the 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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Sergei Treil, Brett Wick. 2007-02-25. Analytic projections, Corona Problem and geometry of holomorphic vector bundles. https://doi.org/10.1090/s0894-0347-08-00611-5

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