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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
DOI: https://doi.org/10.1090/S1061-0022-2011-01186-0
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''.


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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
Email: vasyunin@pdmi.ras.ru

N. K. Nikolski
Affiliation: University Bordeaux 1, France / St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Russia
Email: Nikolai.Nikolski@math.u-bordeaux1.fr

DOI: https://doi.org/10.1090/S1061-0022-2011-01186-0
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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