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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
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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 \Vert\{f_j(z)\}\Vert _E\le 1$, there is a sequence $ \{g_j\}$ of bounded analytic functions with $ \sum _jf_j(z)g_j(z)\equiv 1$ and $ \Vert\{g_j(z)\}\Vert _{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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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

DOI: https://doi.org/10.1090/spmj/1415
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