Extension of matrices with entries in 𝐻^{∞} on coverings of Riemann surfaces of finite type

Brudnyi, A.

doi:10.1090/S1061-0022-10-01101-5

Extension of matrices with entries in $H^{\infty }$ on coverings of Riemann surfaces of finite type
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by A. Brudnyi

St. Petersburg Math. J. 21 (2010), 423-432

DOI: https://doi.org/10.1090/S1061-0022-10-01101-5

Published electronically: February 25, 2010

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Abstract:

The paper continues an earlier work of the author. An extension theorem is proved for matrices with entries in the algebra of bounded holomorphic functions defined on an unbranched covering of a Carathéodory hyperbolic Riemann surface of finite type.

References

David E. Barrett and Jeffrey Diller, A new construction of Riemann surfaces with corona, J. Geom. Anal. 8 (1998), no. 3, 341–347. MR 1707732, DOI 10.1007/BF02921790
Alexander Brudnyi, Projections in the space $H^\infty$ and the corona theorem for subdomains of coverings of finite bordered Riemann surfaces, Ark. Mat. 42 (2004), no. 1, 31–59. MR 2056544, DOI 10.1007/BF02432909
Alexander Brudnyi, Grauert- and Lax-Halmos-type theorems and extension of matrices with entries in $H^\infty$, J. Funct. Anal. 206 (2004), no. 1, 87–108. MR 2024347, DOI 10.1016/S0022-1236(03)00097-1
Alexander Brudnyi, A uniqueness property for $H^\infty$ on coverings of projective manifolds, Michigan Math. J. 51 (2003), no. 3, 503–507. MR 2021004, DOI 10.1307/mmj/1070919556
Alexander Brudnyi, Corona theorem for $H^\infty$ on coverings of Riemann surfaces of finite type, Michigan Math. J. 56 (2008), no. 2, 283–299. MR 2492395, DOI 10.1307/mmj/1224783514
Alexander Brudnyi, Matrix-valued corona theorem for multiply connected domains, Indiana Univ. Math. J. 49 (2000), no. 4, 1405–1410. MR 1836534, DOI 10.1512/iumj.2000.49.1896
Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886
H. Freudenthal, Über dimensionserhöhende stetige Abbildungen, Sitzber. Preus. Akad. Wiss. 5 (1932), 34–38.
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
T. W. Gamelin, Uniform algebras and Jensen measures, London Mathematical Society Lecture Note Series, vol. 32, Cambridge University Press, Cambridge-New York, 1978. MR 521440
John B. Garnett and Peter W. Jones, The corona theorem for Denjoy domains, Acta Math. 155 (1985), no. 1-2, 27–40. MR 793236, DOI 10.1007/BF02392536
Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247
Peter W. Jones and Donald E. Marshall, Critical points of Green’s function, harmonic measure, and the corona problem, Ark. Mat. 23 (1985), no. 2, 281–314. MR 827347, DOI 10.1007/BF02384430
Keiô Nagami, Dimension theory, Pure and Applied Mathematics, Vol. 37, Academic Press, New York-London, 1970. With an appendix by Yukihiro Kodama. MR 0271918
Finnur Lárusson, Holomorphic functions of slow growth on nested covering spaces of compact manifolds, Canad. J. Math. 52 (2000), no. 5, 982–998. MR 1782336, DOI 10.4153/CJM-2000-041-0
V. Ja. Lin, Holomorphic fiberings and multivalued functions of elements of a Banach algebra, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 43–51 (Russian). MR 0318898
Charles N. Moore, The corona theorem for domains whose boundary lies in a smooth curve, Proc. Amer. Math. Soc. 100 (1987), no. 2, 266–270. MR 884464, DOI 10.1090/S0002-9939-1987-0884464-9
E. L. Stout, Bounded holomorphic functions on finite Reimann surfaces, Trans. Amer. Math. Soc. 120 (1965), 255–285. MR 183882, DOI 10.1090/S0002-9947-1965-0183882-4
Vadim Tolokonnikov, Extension problem to an invertible matrix, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1023–1030. MR 1123668, DOI 10.1090/S0002-9939-1993-1123668-X