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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Analytic projections, Corona problem and geometry of holomorphic vector bundles
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by Sergei Treil and Brett D. Wick PDF
J. Amer. Math. Soc. 22 (2009), 55-76 Request permission

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.
References
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Additional Information
  • Sergei Treil
  • Affiliation: Department of Mathematics, Brown University, 151 Thayer Street, Box 1917, Providence, Rhode Island 02912
  • MR Author ID: 232797
  • 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
  • MR Author ID: 766171
  • ORCID: 0000-0003-1890-0608
  • Email: wick@math.sc.edu
  • 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
  • © Copyright 2008 American Mathematical Society
  • 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
  • MathSciNet review: 2449054