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Wolff's theorem on ideals for matrices


Authors: Caleb D. Holloway and Tavan T. Trent
Journal: Proc. Amer. Math. Soc. 143 (2015), 611-620
MSC (2010): Primary 30H05, 30H80
DOI: https://doi.org/10.1090/S0002-9939-2014-12223-4
Published electronically: October 3, 2014
MathSciNet review: 3283648
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Abstract | References | Similar Articles | Additional Information

Abstract: We extend Wolff's theorem concerning ideals on $ H^{\infty }(\mathbb{D}) $ to the matrix case, giving conditions under which an $ H^{\infty } $-solution $ G $ to the equation $ FG = H $ exists for all $ z \in \mathbb{D} $, where $ F $ is an $ m \times \infty $ matrix of functions in $ H^{\infty }(\mathbb{D}) $, and $ H $ is an $ m \times 1 $ vector of such functions. We then examine several useful results.


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  • [1] Mats Andersson, The corona theorem for matrices, Math. Z. 201 (1989), no. 1, 121-130. MR 990193 (90g:30036), https://doi.org/10.1007/BF01161999
  • [2] D. Banjade, Wolff's problem of ideals in the multiplier algebra on Dirichlet space, Complex Analysis and Operator Theory, Jan. 2014, pp. 1-15. ISSN 1661-8254.
  • [3] Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547-559. MR 0141789 (25 #5186)
  • [4] Urban Cegrell, Generalisations of the corona theorem in the unit disc, Proc. Roy. Irish Acad. Sect. A 94 (1994), no. 1, 25-30. MR 1297916 (95k:30069)
  • [5] 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 0222701 (36 #5751)
  • [6] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981. MR 628971 (83g:30037)
  • [7] N. K. Nikolskiĭ, Treatise on the shift operator, Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. MR 827223 (87i:47042)
  • [8] Marvin Rosenblum, A corona theorem for countably many functions, Integral Equations Operator Theory 3 (1980), no. 1, 125-137. MR 570865 (81e:46034), https://doi.org/10.1007/BF01682874
  • [9] J. Ryle and T. T. Trent, A corona theorem for certain subalgebras of $ H^\infty (\mathbb{D})$, Houston J. Math. 37 (2011), no. 4, 1211-1226. MR 2875267 (2012m:46059)
  • [10] 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 (90b:47057)
  • [11] S. Treil, Estimates in the corona theorem and ideals of $ H^\infty $: a problem of T. Wolff, Dedicated to the memory of Thomas H. Wolff. J. Anal. Math. 87 (2002), 481-495. MR 1945294 (2003k:30077), https://doi.org/10.1007/BF02868486
  • [12] Sergei Treil, The problem of ideals of $ H^\infty $: beyond the exponent $ 3/2$, J. Funct. Anal. 253 (2007), no. 1, 220-240. MR 2362422 (2010g:46081), https://doi.org/10.1016/j.jfa.2007.07.018
  • [13] T. Trent, Function theory problems and operator theory, Proceedings of the Topology and Geometry Research Center, TGRC-KOSEF, Vol. 8, Dec. 1997
  • [14] Tavan T. Trent, An estimate for ideals in $ H^\infty (D)$, Integral Equations Operator Theory 53 (2005), no. 4, 573-587. MR 2187440 (2006i:46077), https://doi.org/10.1007/s00020-004-1325-5
  • [15] Tavan Trent and Xinjun Zhang, A matricial corona theorem, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2549-2558. MR 2213732 (2007b:46082), https://doi.org/10.1090/S0002-9939-06-08172-X
  • [16] Tavan T. Trent and Xinjun Zhang, A matricial corona theorem. II, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2845-2854 (electronic). MR 2317961 (2008b:46077), https://doi.org/10.1090/S0002-9939-07-08806-5
  • [17] T. Wolff, A refinement of the corona theorem, in Linear and Complex Analysis Problem Book, V. P. Havin, S. V. Hruscev, and N. K. Nikolski (eds.), Springer-Verlag, Berlin, 1984. MR 0734178 (85k:46001)

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Additional Information

Caleb D. Holloway
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
Email: chollow@uark.edu

Tavan T. Trent
Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487
Email: ttrent@gp.as.ua.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12223-4
Received by editor(s): January 25, 2013
Received by editor(s) in revised form: April 4, 2013
Published electronically: October 3, 2014
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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