Different metrics in the problem of ideals for the algebra 𝐻^{∞}

Kislyakov, S.; Skvortsov, A. A.

doi:10.1090/spmj/1830

Different metrics in the problem of ideals for the algebra $H^\infty$
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by S. V. Kislyakov and A. A. Skvortsov;
Translated by: A. A. Skvortsov

St. Petersburg Math. J. 35 (2024), 815-826

DOI: https://doi.org/10.1090/spmj/1830

Published electronically: December 3, 2024

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Abstract:

Starting with the 1970s, significant attention has been paid to estimates of solutions of equations arising in the corona theorem and the associated so-called ideal problem. Naturally, the question emerged about the metrics in which such estimates should be sought. In the case of the corona theorem, the answer to this question is known: it is practically always possible to pass from estimates in one reasonable metric to estimates in any other. In this article, something similar is proved for the ideal problem. It should be noted that in the case of the ideal problem, formulations themselves with different metrics are not always obvious.

The article concludes with several applications to the operator corona theorem and ideal problem.

The proofs of the main results heavily rely on fixed point theorems.

References

A. V. Buhvalov, Hardy spaces of vector-valued functions, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 65 (1976), 5–16, 203 (Russian, with English summary). Investigations on linear operators and theory of functions, VII. MR 493312
I. K. Zlotnikov, The problem of ideals in the algebra $H^\infty$ in the case of certain spaces of sequences, Algebra i Analiz 29 (2017), no. 5, 51–67 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 29 (2018), no. 5, 749–759. MR 3724638, DOI 10.1090/spmj/1514
L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
S. V. Kislyakov and D. V. Rutskiĭ, Some remarks on the corona theorem, Algebra i Analiz 24 (2012), no. 2, 171–191 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 24 (2013), no. 2, 313–326. MR 3013331, DOI 10.1090/S1061-0022-2013-01240-4
S. V. Kislyakov and A. S. Skvortsov, Fixed point theorems and Hardy classes, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 512 (2022), no. Issledovaniya po Lineĭnym Operatoram i Teorii Funktsiĭ. 50, 95–115 (Russian, with English summary); English transl., J. Math. Sci. (N.Y.) 282 (2024), no. 4, 516–529. MR 4508361
Paul Koosis, Introduction to $H_{p}$ spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451
G. Ja. Lozanovskiĭ, Certain Banach lattices, Sibirsk. Mat. Ž. 10 (1969), 584–599 (Russian). MR 241949
I. I. Privalov, Graničnye svoĭstva analitičeskih funkciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR 47765
V. A. Tolokonnikov, Estimates in the Carleson corona theorem, ideals of the algebra $H^{\infty }$, a problem of Sz.-Nagy, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 178–198, 267 (Russian, with English summary). Investigations on linear operators and the theory of functions, XI. MR 629839
S. R. Treil′, Angles between co-invariant subspaces, and the operator corona problem. The Szőkefalvi-Nagy problem, Dokl. Akad. Nauk SSSR 302 (1988), no. 5, 1063–1068 (Russian); English transl., Soviet Math. Dokl. 38 (1989), no. 2, 394–399. MR 981054
Ky. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121–126. MR 47317, DOI 10.1073/pnas.38.2.121
Paul A. Fuhrmann, On the corona theorem and its application to spectral problems in Hilbert space, Trans. Amer. Math. Soc. 132 (1968), 55–66. MR 222701, DOI 10.1090/S0002-9947-1968-0222701-7
Sergei Treil and Brett D. Wick, The matrix-valued $H^p$ corona problem in the disk and polydisk, J. Funct. Anal. 226 (2005), no. 1, 138–172. MR 2158178, DOI 10.1016/j.jfa.2005.04.010
Sergei Treil, The problem of ideals of $H^\infty$: beyond the exponent $3/2$, J. Funct. Anal. 253 (2007), no. 1, 220–240. MR 2362422, DOI 10.1016/j.jfa.2007.07.018
Dmitry V. Rutsky, Corona problem with data in ideal spaces of sequences, Arch. Math. (Basel) 108 (2017), no. 6, 609–619. MR 3646952, DOI 10.1007/s00013-017-1045-0