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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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

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