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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Invertibility threshold for $ H^\infty$ trace algebras, and effective matrix inversions

Authors: V. I. Vasyunin and N. K. Nikolski
Translated by: the authors
Original publication: Algebra i Analiz, tom 23 (2011), nomer 1.
Journal: St. Petersburg Math. J. 23 (2012), 57-73
MSC (2010): Primary 47L80; Secondary 30H05
Published electronically: November 7, 2011
MathSciNet review: 2760148
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Abstract | References | Similar Articles | Additional Information

Abstract: Given $ \delta$, $ 0<\delta<1$, a Blaschke sequence $ \sigma=\{\lambda_j\}$ is constructed such that every function $ f\in H^\infty$ satisfying $ \delta<\delta_f=\inf_{\lambda\in\sigma}\vert f(\lambda)\vert\le\Vert f\Vert _\infty\le1$ is invertible in the trace algebra $ H^\infty\vert\sigma$ (with a norm estimate of the inverse depending on $ \delta_f$ only), but there exists $ f$ with $ \delta=\delta_f\le\Vert f\Vert _\infty\le1$ that is not. As an application, a counterexample to a stronger form of the Bourgain-Tzafriri restricted invertibility conjecture for bounded operators is exhibited, where an ``orthogonal (or unconditional) basis'' is replaced by a ``summation block orthogonal basis''.

References [Enhancements On Off] (What's this?)

  • [ARCC] ARCC Workshop: The Kadison-Singer Problem,
  • [BTz] J. Bourgain and L. Tzafriri, Invertibility of ``large'' submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel J. Math. 57 (1987), no. 2, 137-224. MR 0890420 (89a:46035)
  • [CCLV] P. G. Casazza, O. Christensen, A. M. Lindner, and R. Vershynin, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc. 133 (2005), 1025-1033. MR 2117203 (2006a:46024)
  • [CT] P. G. Casazza and J. C. Tremain, The Kadison-Singer problem in mathematics and engineering, Proc. Nat. Acad. Sci. USA 103 (2006), 2032-2039. MR 2204073 (2006j:46074)
  • [Gar] J. B. Garnett, Bounded analytic functions, Pure Appl. Math., vol. 96, Acad. Press, New York-London, 1981. MR 0628971 (83g:30037)
  • [GMN] P. Gorkin, R. Mortini, and N. Nikolski, Norm controlled inversions and a corona theorem for $ H^\infty$-quotient algebras, J. Funct. Anal. 255 (2008), 854-876. MR 2433955 (2009h:30061)
  • [KS] R. Kadison and I. Singer, Extensions of pure states, Amer. J. Math. 81 (1959), 383-400. MR 0123922 (23:A1243)
  • [Nik1] N. Nikol'skiĭ, Lectures on the shift operator, Nauka, Moscow, 1980; English transl., Treatise on the shift operator, Grundlehren Math. Wiss., Bd. 273, Springer-Verlag, Berlin, 1986. MR 0575166 (82i:47013); MR 0827223 (87i:47042)
  • [Nik2] -, Operators, functions, and systems. Vol. 1, 2, Math. Surveys Monogr., vol. 92, 93, Amer. Math. Soc., Providence, RI, 2002. MR 1864396 (2003i:47001a); MR 1892647 (2003i:47001b)
  • [SS] D. A. Spielman and N. Srivastava, An elementary proof of the restricted invertibility theorem, arXiv:0911.1114v3 (February 2, 2010).

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Additional Information

V. I. Vasyunin
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, Petersburg 191023, Russia

N. K. Nikolski
Affiliation: University Bordeaux 1, France / St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Russia

Keywords: Effective inversions, $H^\infty$ trace algebra, invisible spectrum, critical constant, interpolation Blaschke product, Bourgain--Tzafriri restricted invertibility conjecture
Received by editor(s): September 12, 2010
Published electronically: November 7, 2011
Additional Notes: V. Vasyunin’s research was supported in part by RFBR (grant no. 08-01-00723)
N. Nikolski’s research was partially supported by the French ANR Projects DYNOP and FRAB
Dedicated: Dedicated to the memory of M. Sh. Birman, from whom both of us learned a lot (and not only in mathematics)
Article copyright: © Copyright 2011 American Mathematical Society

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