A matricial corona theorem

Trent, Tavan; Zhang, Xinjun

doi:10.1090/S0002-9939-06-08172-X

A matricial corona theorem
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by Tavan Trent and Xinjun Zhang PDF

Proc. Amer. Math. Soc. 134 (2006), 2549-2558 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.