Invertibility threshold for 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 , , a Blaschke sequence is constructed such that every function satisfying is invertible in the trace algebra (with a norm estimate of the inverse depending on only), but there exists with 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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Nikolai K. Nikolski,*Operators, functions, and systems: an easy reading. Vol. 2*, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. MR**1892647****[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

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:
http://dx.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