A matricial corona theorem
HTML articles powered by AMS MathViewer
- by Tavan Trent and Xinjun Zhang
- Proc. Amer. Math. Soc. 134 (2006), 2549-2558
- DOI: https://doi.org/10.1090/S0002-9939-06-08172-X
- Published electronically: April 7, 2006
- PDF | Request permission
Abstract:
We show that a usual corona-type theorem on a space of functions automatically extends to a matrix version.References
- G. Birkhoff and S. Mac Lane, Algebra, MacMillan, Toronto, 1971.
- 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
- Stephen D. Fisher, Function theory on planar domains, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A second course in complex analysis; A Wiley-Interscience Publication. MR 694693
- Frank Forelli, Bounded holomorphic functions and projections, Illinois J. Math. 10 (1966), 367–380. MR 193534
- Paul A. Fuhrmann, On the corona theorem and its application to spectral problems in Hilbert space, Trans. Amer. Math. Soc. 132 (1968), 55–66. MR 222701, DOI 10.1090/S0002-9947-1968-0222701-7
- Artur Nicolau, The corona property for bounded analytic functions in some Besov spaces, Proc. Amer. Math. Soc. 110 (1990), no. 1, 135–140. MR 1017007, DOI 10.1090/S0002-9939-1990-1017007-X
- N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1
- Marvin Rosenblum, A corona theorem for countably many functions, Integral Equations Operator Theory 3 (1980), no. 1, 125–137. MR 570865, DOI 10.1007/BF01682874
- 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
- V. A. Tolokonnikov, Estimates in Carleson’s corona theorem and finitely generated ideals of the algebra $H^{\infty }$, Funktsional. Anal. i Prilozhen. 14 (1980), no. 4, 85–86 (Russian). MR 595742
- S. R. Treil′, Angles between co-invariant subspaces, and the operator corona problem. The Szőkefalvi-Nagy problem, Dokl. Akad. Nauk SSSR 302 (1988), no. 5, 1063–1068 (Russian); English transl., Soviet Math. Dokl. 38 (1989), no. 2, 394–399. MR 981054
- Tavan T. Trent, A corona theorem for multipliers on Dirichlet space, Integral Equations Operator Theory 49 (2004), no. 1, 123–139. MR 2057771, DOI 10.1007/s00020-002-1196-6
- Tavan T. Trent, A new estimate for the vector valued corona problem, J. Funct. Anal. 189 (2002), no. 1, 267–282. MR 1887635, DOI 10.1006/jfan.2001.3842
- —, An $H^p$-corona theorem on the bidisk for infinitely many functions, submitted.
- X. Zhang, A matrix version of corona theorem for algebras of functions on reproducing kernel Hilbert spaces, Ph.D. dissertation, The University of Alabama, Tuscaloosa, AL, August 2004.
Bibliographic Information
- Tavan Trent
- Affiliation: Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350
- Email: ttrent@gp.as.ua.edu
- Xinjun Zhang
- Affiliation: Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350
- Email: zhang010@bama.ua.edu
- Received by editor(s): September 8, 2004
- Received by editor(s) in revised form: January 13, 2005
- Published electronically: April 7, 2006
- Additional Notes: This work was partially supported by NSF Grant DMS-0400307.
- Communicated by: Joseph A. Ball
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 2549-2558
- MSC (2000): Primary 32A65, 46J20
- DOI: https://doi.org/10.1090/S0002-9939-06-08172-X
- MathSciNet review: 2213732