Analytic projections, Corona Problem and geometry of holomorphic vector bundles

Details Analytic projections, Corona Problem and geometry of holomorphic vector bundles Analytic projections, Corona Problem and geometry of holomorphic vector bundles

2007-02-25

arxiv.org/abs/math/0702756v2

J. Amer. Math. Soc. 22 (2009), no. 1, 55-76

Mathematics

Classical Analysis and ODEs

24 pages. Accepted by the "Journal of AMS". In this version typos are corrected and the presentation is improved according to

Scientific paper

10.1090/S0894-0347-08-00611-5

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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