Abstract
Let E be a Banach lattice on \({\mathbb {Z}}\) with order continuous norm. We show that for any function \(f = \{f_j\}_{j \in {\mathbb {Z}}}\) from the Hardy space \(\mathrm H_{\infty }\left( E \right) \) such that \(\delta \leqslant \Vert f (z)\Vert _E \leqslant 1\) for all z from the unit disk \({\mathbb {D}}\) there exists some solution \(g = \{g_j\}_{j \in {\mathbb {Z}}} \in \mathrm H_{\infty }\left( E' \right) \), \(\Vert g\Vert _{\mathrm H_{\infty }\left( E' \right) } \leqslant C_\delta \) of the Bézout equation \(\sum _j f_j g_j = 1\), also known as the vector-valued corona problem with data in \(\mathrm H_{\infty }\left( E \right) \).
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Acknowledgements
The author is grateful to S. V. Kislyakov for stimulating discussions and a surprising conjecture that eventually became the statement of Theorem 2, and to the referee for thorough and helpful remarks.
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Rutsky, D.V. Corona problem with data in ideal spaces of sequences. Arch. Math. 108, 609–619 (2017). https://doi.org/10.1007/s00013-017-1045-0
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DOI: https://doi.org/10.1007/s00013-017-1045-0