Analytic projections, Corona Problem and geometry of holomorphic vector bundles
arXiv:math/0702756 · doi:10.1090/S0894-0347-08-00611-5
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.
24 pages. Accepted by the "Journal of AMS". In this version typos are corrected and the presentation is improved according to the suggestions of the referee