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St. Petersburg Mathematical Journal

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



Problem of ideals in the algebra $ H^{\infty}$ for some spaces of sequences

Author: I. K. Zlotnikov
Translated by: the author
Original publication: Algebra i Analiz, tom 29 (2017), nomer 5.
Journal: St. Petersburg Math. J. 29 (2018), 749-759
MSC (2010): Primary 30H80
Published electronically: July 26, 2018
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Abstract: Metric aspects of the problem of ideals are studied. Let $ h$ be a function in the class $ H^{\infty }(\mathbb{D})$ and $ f$ a vector-valued function in the class $ H^{\infty }(\mathbb{D};E)$, i.e., $ f$ takes values in some lattice of sequences $ E$. Suppose that $ \vert h(z)\vert \le \Vert f(z)\Vert^{\alpha }_{E} \le 1$ for some parameter $ \alpha $. The task is to find a function $ g$ in $ H^{\infty }(\mathbb{D};E')$, where $ E'$ is the order dual of $ E$, such that $ \sum f_j g_j = h$. Also it is necessary to control the value of $ \Vert g\Vert _{H^{\infty }(E')}$. The classical case with $ E = l^2$ was investigated by V. A. Tolokonnikov in 1981. Recently, the author managed to obtain a similar result for the space $ E = l^1$. In this paper it is shown that the problem of ideals can be solved for any $ q$-concave Banach lattice $ E$ with finite $ q$; in particular, $ E=l^p$ with $ p \in [1,\infty )$ fits.

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Additional Information

I. K. Zlotnikov
Affiliation: Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O. 29B, St.Petersburg 199178, Russia; St. Petersburg Branch, Steklov Institute of Mathematics, Russian Academy of Sciences, 27 Fontanka, St. Petersburg 191023, Russia

Keywords: Problem of ideals, corona theorem, Kakutani fixed point theorem
Received by editor(s): May 1, 2017
Published electronically: July 26, 2018
Additional Notes: Supported by RFBR (grant no. 17-01-00607) and by “Native towns”, a social investment program of PJSC “Gazprom Neft”
Article copyright: © Copyright 2018 American Mathematical Society

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