×

A new estimate for the vector valued corona problem. (English) Zbl 1004.30026

In this paper the author gives an improved estimate in the vector valued corona theorem for the unit disk. His proof splits into two parts: first, he resurrects the operator corona theorem from the 1970s, and then he uses a Hilbert space argument, based on a version of Wolff’s ideas. Since he uses Hilbert space methods, the Riesz representation theorem replaces the usual \(\bar\partial\)-techniques. Then the author considers the operator version of the corona problem on the unit disk \(D\) and gives an improvement of Vasyunin’s estimate. He gets estimates for analytic solutions, \(G(z)\), of \(F(z)G(z)=I\) for all \(z\in D\) of the form \[ \sup\limits_{z\in D}||G(z)||_{B(\mathbb{C}^m,\mathbb{C}^n)}\leq 2[2\sqrt{e}+ 2\sqrt{2}e]\frac 1{\varepsilon^{n+1}}\ln\left(\frac 1{\varepsilon^{2n}}\right) . \] Since estimates independent of \(m\), the case \(m=\infty\) follows from a routine argument.
Reviewer: K.Malyutin (Sumy)

MSC:

30D55 \(H^p\)-classes (MSC2000)
32A38 Algebras of holomorphic functions of several complex variables
47C10 Linear operators in \({}^*\)-algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agler, J.; McCarthy, J. W., Nevanlinna-Pick interpolation on the bidisk, J. Reine Angew. Math., 506, 191-204 (1999) · Zbl 0919.32003
[2] Arveson, W. B., Interpolation problems in nest algebras, J. Funct. Anal., 3, 208-233 (1975) · Zbl 0309.46053
[3] Ball, J. A.; Li, W. S.; Timotin, D.; Trent, T. T., A commutant lifting theorem on the polydisk: Interpolation problems for the bidisc, Indiana Univ. J., 48, 653-675 (1999) · Zbl 0953.32010
[4] Ball, J. A.; Trent, T., Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables, J. Funct. Anal., 157, 1-61 (1998) · Zbl 0914.47020
[5] Carleson, L., Interpolation by bounded analytic functions and the corona problem, Ann. of Math., 76, 547-559 (1962) · Zbl 0112.29702
[6] Fleming, W., Functions of Several Variables (1965), Addison-Wesley: Addison-Wesley Atlanta · Zbl 0136.34301
[7] Fuhrmann, P. A., On the corona theorem and its application to spectral problems in Hilbert space, Trans. Amer. Math. Soc., 132, 55-66 (1968) · Zbl 0187.38002
[8] Garnett, J. B., Bounded Analytic Functions (1981), Academic Press: Academic Press New York · Zbl 0469.30024
[9] Helton, J. W., Optimization over \(H^∞\) and the Toeplitz corona theorem, J. Operator Theory, 15, 359-375 (1986) · Zbl 0605.46043
[10] Li, S.-Y., Corona problems of several complex variables, Madison Symposium of Complex Analysis. Madison Symposium of Complex Analysis, Contemporary Mathematics, 137 (1991), Amer. Math. Soc: Amer. Math. Soc Providence
[11] Lin, K. C., \(H^p\) corona theorem for the polydisk, Trans. Amer. Math. Soc., 341, 371-375 (1994) · Zbl 0798.32005
[12] Katsoulis, E. G.; Moore, R. L.; Trent, T. T., Interpolation in nest algebras and applications to operator corona theorems, J. Operator Theory, 29, 115-123 (1993) · Zbl 0813.47053
[13] Nikolskii, N. K., Treatise on the Shift Operator (1985), Springer-Verlag: Springer-Verlag New York
[14] Rosenblum, M., A corona theorem for countably many functions, Integral Equations Operator Theory, 3, 125-137 (1980) · Zbl 0452.46032
[15] Schubert, C. F., Corona theorem as operator theorem, Proc. Amer. Math. Soc., 69, 73-76 (1978) · Zbl 0387.47023
[16] Sz.-Nagy, B.; Foias, C., On contractions similar to Toeplitz operators, Ann. Acad. Sci. Fenn. Ser. A I Math., 2, 553-564 (1976) · Zbl 0324.47005
[17] Tolokonnikov, V. A., Estimates in Carleson’s corona theorem and finitely generated ideals in the algebra \(H^∞\), Functional. Anal. I Prilozhen, 14, 85-86 (1980)
[18] Treil, S. R., Angles between coinvariant subspaces and an operator-valued corona problem, a question of Szokefalvi-Nagy, Soviet Math. Dokl., 38, 394-399 (1989) · Zbl 0687.47004
[19] Trent, T. T., Function theory problems and operator theory, Proceedings of the Topology and Geometry Research Center. Proceedings of the Topology and Geometry Research Center, TGRC-KOSEF, 8 (1997), p. 47-60
[20] A. Uchiyama, Corona theorems for countably many functions and estimates for their solutions, preprint, 1980.; A. Uchiyama, Corona theorems for countably many functions and estimates for their solutions, preprint, 1980.
[21] Havin, V. P.; Hruscev, S. V.; Nikolskii, N. K., Linear and Complex Analysis Problem Book. Linear and Complex Analysis Problem Book, Lecture Notes in Mathematics, 1043 (1984), Springer-Verlag: Springer-Verlag New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.