Wolff’s theorem on ideals for matrices

Holloway, Caleb; Trent, Tavan

doi:10.1090/S0002-9939-2014-12223-4

Wolff’s theorem on ideals for matrices
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by Caleb D. Holloway and Tavan T. Trent

Proc. Amer. Math. Soc. 143 (2015), 611-620

DOI: https://doi.org/10.1090/S0002-9939-2014-12223-4

Published electronically: October 3, 2014

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Abstract:

We extend Wolff’s theorem concerning ideals on $H^{\infty }(\mathbb {D})$ to the matrix case, giving conditions under which an $H^{\infty }$-solution $G$ to the equation $FG = H$ exists for all $z \in \mathbb {D}$, where $F$ is an $m \times \infty$ matrix of functions in $H^{\infty }(\mathbb {D})$, and $H$ is an $m \times 1$ vector of such functions. We then examine several useful results.

References

Mats Andersson, The corona theorem for matrices, Math. Z. 201 (1989), no. 1, 121–130. MR 990193, DOI 10.1007/BF01161999
D. Banjade, Wolff’s problem of ideals in the multiplier algebra on Dirichlet space, Complex Analysis and Operator Theory, Jan. 2014, pp. 1–15. ISSN 1661-8254.
Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
Urban Cegrell, Generalisations of the corona theorem in the unit disc, Proc. Roy. Irish Acad. Sect. A 94 (1994), no. 1, 25–30. MR 1297916
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
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1
Marvin Rosenblum, A corona theorem for countably many functions, Integral Equations Operator Theory 3 (1980), no. 1, 125–137. MR 570865, DOI 10.1007/BF01682874
J. Ryle and T. T. Trent, A corona theorem for certain subalgebras of $H^\infty (\Bbb D)$, Houston J. Math. 37 (2011), no. 4, 1211–1226. MR 2875267
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
S. Treil, Estimates in the corona theorem and ideals of $H^\infty$: a problem of T. Wolff, J. Anal. Math. 87 (2002), 481–495. Dedicated to the memory of Thomas H. Wolff. MR 1945294, DOI 10.1007/BF02868486
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
T. Trent, Function theory problems and operator theory, Proceedings of the Topology and Geometry Research Center, TGRC-KOSEF, Vol. 8, Dec. 1997
Tavan T. Trent, An estimate for ideals in $H^\infty (D)$, Integral Equations Operator Theory 53 (2005), no. 4, 573–587. MR 2187440, DOI 10.1007/s00020-004-1325-5
Tavan Trent and Xinjun Zhang, A matricial corona theorem, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2549–2558. MR 2213732, DOI 10.1090/S0002-9939-06-08172-X
Tavan T. Trent and Xinjun Zhang, A matricial corona theorem. II, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2845–2854. MR 2317961, DOI 10.1090/S0002-9939-07-08806-5
V. P. Havin, S. V. Hruščëv, and N. K. Nikol′skiĭ (eds.), Linear and complex analysis problem book, Lecture Notes in Mathematics, vol. 1043, Springer-Verlag, Berlin, 1984. 199 research problems. MR 734178, DOI 10.1007/BFb0072183