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St. Petersburg Mathematical Journal

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

 
 

 

Corona theorem and interpolation


Author: S. V. Kislyakov
Translated by: the author
Original publication: Algebra i Analiz, tom 27 (2015), nomer 5.
Journal: St. Petersburg Math. J. 27 (2016), 757-764
MSC (2010): Primary 30H80
DOI: https://doi.org/10.1090/spmj/1415
Published electronically: July 26, 2016
MathSciNet review: 3582942
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $E$ be a Banach ideal space of sequences and $E’$ its order dual. By definition, $E$ verifies the corona theorem if for arbitrary bounded functions $f_j$ analytic in the unit disk $\mathbb {D}$ and such that $0<\delta \le \|\{f_j(z)\}\|_E\le 1$, there is a sequence $\{g_j\}$ of bounded analytic functions with $\sum _jf_j(z)g_j(z)\equiv 1$ and $\|\{g_j(z)\}\|_{E’}\le C(\delta )$, $z\in \mathbb {D}$. It is shown that the spaces $\ell ^p$, $1\le p<\infty$, and some more general Banach lattices verify the corona theorem.


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References
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Additional Information

S. V. Kislyakov
Affiliation: St. Petersburg Branch, Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia; Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Staryi Petergof, St. Petersburg 198504, Russia
Email: skis@pdmi.ras.ru

Keywords: Corona theorem, lattice of measurable functions, $\mathrm {BMO}$-regularity
Received by editor(s): June 30, 2015
Published electronically: July 26, 2016
Additional Notes: Supported by RFBR, grant no. 14-01-00198
Article copyright: © Copyright 2016 American Mathematical Society