Advertisement

Ideal Membership in \(H^\infty \): Toeplitz Corona Approach

  • Michael Hartz
  • Brett D. Wick
Article
  • 36 Downloads

Abstract

We study the ideal membership problem in \(H^\infty \) on the unit disc. Thus, given functions \(f,f_1,\ldots ,f_n\) in \(H^\infty \), we seek sufficient conditions on the size of f in order for f to belong to the ideal of \(H^\infty \) generated by \(f_1,\ldots ,f_n\). We provide a different proof of a theorem of Treil, which gives the sharpest known sufficient condition. To this end, we solve a closely related problem in the Hilbert space \(H^2\), which is equivalent to the ideal membership problem by the Nevanlinna–Pick property of \(H^2\).

Keywords

Corona problem Ideal membership Carleson measure Nevanlinna–Pick 

Mathematics Subject Classification

Primary 30H05 Secondary 46J20 30H80 

References

  1. 1.
    Agler, J., McCarthy, J.E.: Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Providence, RI (2002)zbMATHGoogle Scholar
  2. 2.
    Carleson, L.: Interpolations by bounded analytic functions and the corona problem, Ann. Math. (2) 76, 547–559 (1962)Google Scholar
  3. 3.
    Cegrell, U.: A generalization of the corona theorem in the unit disc. Math. Z. 203(2), 255–261 (1990)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cegrell, U.: Generalisations of the corona theorem in the unit disc. Proc. Roy. Irish Acad. Sect. A 94(1), 25–30 (1994)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Costea, Ş., Sawyer, E.T., Wick, B.D.: The corona theorem for the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in \(\mathbb{C}^n\). Anal. PDE 4(4), 499–550 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gamelin, T.W.: Wolff’s proof of the corona theorem. Israel J. Math. 37(1–2), 113–119 (1980)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Garnett, J.B.: Bounded analytic functions, first ed., Graduate Texts in Mathematics, vol. 236, Springer, New York (2007)Google Scholar
  8. 8.
    Kaashoek, M.A., Rovnyak, J.: On the preceding paper by R. B. Leech. Integral Equ. Oper. Theory 78(1), 75–77 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Robert, B.: Leech. Factorization of analytic functions and operator inequalities, Integral Equations Operator Theory 78(1), 71–73 (2014)Google Scholar
  10. 10.
    Lin, K.-C.: On the constants in the corona theorem and the ideals of \(H^\infty \). Houston J. Math. 19(1), 97–106 (1993)MathSciNetGoogle Scholar
  11. 11.
    Pau, J.: On a generalized corona problem on the unit disc. Proc. Am. Math. Soc. 133(1), 167–174 (2005). (electronic)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Rao, K.V.R.: On a generalized corona problem. Journal d’Analyse Mathématique 18(1), 277–278 (1967)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Treil, S.: Estimates in the corona theorem and ideals of \(H^\infty \): a problem of T. Wolff, J. Anal. Math. 87, 481–495. Dedicated to the memory of Thomas H, Wolff (2002)Google Scholar
  14. 14.
    Treil, S.: The problem of ideals of \(H^\infty \): beyond the exponent \(3/2\). J. Funct. Anal. 253(1), 220–240 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Treil, S., Wick, B.D.: The matrix-valued \(H^p\) corona problem in the disk and polydisk. J. Funct. Anal. 226(1), 138–172 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Tavan, T.: Trent, An estimate for ideals in \(H^\infty (D)\). Integral Equ. Oper. Theory 53(4), 573–587 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsWashington University in St. LouisSt. LouisUSA

Personalised recommendations