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Analytic projections, Corona problem and geometry of holomorphic vector bundles


Authors: Sergei Treil and Brett D. Wick
Journal: J. Amer. Math. Soc. 22 (2009), 55-76
MSC (2000): Primary 30D55; Secondary 46J15, 46J20
DOI: https://doi.org/10.1090/S0894-0347-08-00611-5
Published electronically: July 31, 2008
MathSciNet review: 2449054
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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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Additional Information

Sergei Treil
Affiliation: Department of Mathematics, Brown University, 151 Thayer Street, Box 1917, Providence, Rhode Island 02912
Email: treil@math.brown.edu

Brett D. Wick
Affiliation: Department of Mathematics, University of South Carolina, LeConte College, 1523 Greene Street, Columbia, South Carolina 29208
Email: wick@math.sc.edu

DOI: https://doi.org/10.1090/S0894-0347-08-00611-5
Keywords: Corona Theorem, analytic projections, Nikolski's lemma
Received by editor(s): January 14, 2006
Published electronically: July 31, 2008
Additional Notes: The work of the first author was supported by the National Science Foundation under Grant DMS-0501065
Article copyright: © Copyright 2008 American Mathematical Society

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