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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
MathSciNet review: 3724638
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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 $|h(z)| \le \|f(z)\|^{\alpha }_{E} \le 1$ 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 $\|g\|_{H^{\infty }(E’)}$. 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