Bounding smooth solutions of Bezout equations

  • Nikolai Nikolski

Abstract

Given data $f=(f_1,f_2,\dots ,f_n)$ in the holomorphic part $ A= F_+$ of a symmetric Banach\slash topological algebra $ F$ on the unit circle $\mathbb{T}$, we estimate solutions $ g_j\in A$ to the corresponding Bezout equation $\sum _{j=1}^ng_jf_j=1$ in terms of the lower spectral parameter δ, $0< \delta \leq |f(z)|$, and an inversion controlling function $c_1(\delta ,F)$ for the algebra $F$. A scheme developed issues from an analysis of the famous Uchiyama-Wolff proof to the Carleson corona theorem and includes examples of algebras of “smooth” functions, as Beurling-Sobolev, Lipschitz, or Wiener-Dirichlet algebras. There is no real “corona problem” in this setting, the issue is in the growth rate of the upper bound for $\|g\|_{A^n}$ as $\delta \to 0$ and in numerical values of the quantities that occur, which are determined as accurately as possible.

References

Aleksandrov, A. B., Norm of the Hilbert transform in a space of Hölder functions, Funkcional. Anal. i Priložen. 9 (1975), no. 2, 1–4.

Aleman, A. and Dahlner, A., Uniform spectral radius and compact Gelfand transform, Studia Math. 172 (2006), no. 1, 25–46. https://doi.org/10.4064/sm172-1-2

Douglas, J., Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263–321. https://doi.org/10.2307/1989472

Douglas, R. G., Krantz, S. G., Sawyer, E. T., Treil, S., and Wicks, B. D., A history of the corona problem, in “The corona problem”, Fields Inst. Commun., vol. 72, Springer, New York, 2014, pp. 1--29. https://doi.org/10.1007/978-1-4939-1255-1_1

Duren, P. L., Theory of $H^p$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970.

El-Fallah, O. and Ezzaaraoui, A., Majorations uniformes de normes d'inverses dans les algèbres de Beurling, J. London Math. Soc. (2) 65 (2002), no. 3, 705–719. https://doi.org/10.1112/S0024610701003088

El-Fallah, O., Nikolski, N. K., and Zarrabi, M., Estimates for resolvents in Beurling-Sobolev algebras, Algebra i Analiz 10 (1998), no. 6, 1–92.

El-Fallah, O. and Zarrabi, M., Estimations des solutions de l'équation de Bézout dans les algèbres de Beurling analytiques, Math. Scand. 96 (2005), no. 2, 307–319. https://doi.org/10.7146/math.scand.a-14959

Garnett, J. B., Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc., New York-London, 1981.

Nikolski, N. K., Basisness and unicellularity of weighted shift operators, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1123–1137.

Nikolski, N. K., Treatise on the shift operator. Spectral function theory, Grundlehren der Mathematischen Wissenschaften, vol. 273, Springer-Verlag, Berlin, 1986. https://doi.org/10.1007/978-3-642-70151-1

Nikolski, N. K., In search of the invisible spectrum, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 6, 1925–1998.

Nikolski, N. K., The problem of efficient inversions and Bezout equations, in “Twentieth century harmonic analysis—a celebration (Il Ciocco, 2000)'', NATO Sci. Ser. II Math. Phys. Chem., vol. 33, Kluwer Acad. Publ., Dordrecht, 2001, pp. 235--269.

Nikolski, N. K., Operators, functions, and systems: an easy reading. Vol. 2: Model operators and systems, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002.

Nikolski, N. K., Matrices et opérateurs de Toeplitz, Mathématiques en devenir, Calvage et Mounet, 2017.

Olofsson, A., An extremal problem in Banach algebras, Studia Math. 145 (2001), no. 3, 255–264. https://doi.org/10.4064/sm145-3-5

Shamoyan, F. A., Applications of Dzhrbashyan integral representations to some problems of analysis, Dokl. Akad. Nauk SSSR 261 (1981), no. 3, 557–561.

Vinogradov, S. A., Properties of multipliers of integrals of Cauchy-Stieltjes type, and some problems of factorization of analytic functions, in “Mathematical programming and related questions (Proc. Seventh Winter School, Drogobych, 1974), Theory of functions and functional analysis”, Central Èkonom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 5--39.

Published
2017-09-22
How to Cite
Nikolski, N. (2017). Bounding smooth solutions of Bezout equations. MATHEMATICA SCANDINAVICA, 121(1), 121-143. https://doi.org/10.7146/math.scand.a-26387
Section
Articles