Corona theorem and interpolation

Kislyakov, S.

doi:10.1090/spmj/1415

Corona theorem and interpolation
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by S. V. Kislyakov
Translated by: the author

St. Petersburg Math. J. 27 (2016), 757-764

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 $0<\delta \le \|\{f_j(z)\}\|_E\le 1$, 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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