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Regular version of the site

Article

Problem of ideals in the algebra H∞for some spaces of sequences

St Petersburg Mathematical Journal. 2018. Vol. 29. No. 5. P. 749-759.
Zlotnikov I.K.

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., takes values in some lattice of sequences E. Suppose that |h(z)| \le \|f(z)\|_{E}^{\alpha} \le 1 for some parameter \alpha. The task is to find g a function 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.