Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Estimates in the Carleson corona theorem, ideals of the algebra H, a problem of S.-Nagy

  • 32 Accesses

  • 3 Citations


Let E1, E2, be Hilbert spaces, H(E1,E2) be the space of functions, bounded and analytic in the disk D, with values in the space of bounded linear operators from E1 to E2. Estimates are investigated for a solution of the problem of S.-Nagy of finding a left inverse element for a function F, FεH(E1,E2). For dim E1=1 this problem is a generalization of the corona problem. Let Cn(δ)= sup¦∶FεH(E1,E2),dim E1=n, ¦F¦⩽1, ¦F(z)a¦2⩾δ¦a¦2(zεD,aεE1);Gε H(E2,E1) is a function of minimal norm for which

. Then

where an, Cn are constants depending only on n. The behavior of the function C1 as δ→1 is described. Other results are obtained also.

This is a preview of subscription content, log in to check access.

Literature cited

  1. 1.

    P. Koosis, “Introduction to Hp spaces,” London Math. Soc. Lect. Notes Ser. No. 40 (1980).

  2. 2.

    N. K. Nikol'skii, Lectures on the Shift Operator [in Russian], Nauka, Moscow (1980).

  3. 3.

    M. Rosenblum, “A corona theorem for countably many functions,” Integral Equations and Operator Theory,3, No. 1, 125–137 (1980).

  4. 4.

    B. S. Nagy, “A problem on operator valued bounded analytic functions,” Zap. Nauch. Sem. Leningr. Otd. Mat. Inst.,81, 99 (1978).

  5. 5.

    R. Bellman, Introduction to Matrix Theory [Russian translation], Nauka, Moscow (1969).

  6. 6.

    J. P. Rosay, “Une equivalence on corona probleme dans ℂn et un probleme d'ideal dans H,” J. Funct. Anal.,7, No. 1, 71–84 (1971).

  7. 7.

    S. A. Vinogradov and V. P. Khavin, “Free interpolation in H and in certain other classes of functions,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,47, 15–54 (1974).

  8. 8.

    R. G. Douglas, “Banach algebra techniques in the theory of Toeplitz operators,” CBMS Regional Conference,15, American Math. Soc., Providence, R. I. (1973).

  9. 9.

    F. Forelli, “Bounded holomorphic functions and projections,” Ill. J. Math.,10, 367–380 (1966).

  10. 10.

    D. Sarason, “Function theory on the unit circle,” Lect. Notes, Virginia (1978).

  11. 11.

    R. Rao, “On a generalized corona problem,” J. Anal. Math.,18, 277–278 (1967).

  12. 12.

    C. F. Shubert, “The corona theorem as an operator theorem,” Proc. Am. Math. Soc.,69, No. 1, 73–76 (1978).

  13. 13.

    B. S. Nagy and C. Foias, “On contractions similar to isometries and Toeplitz operators,” Ann. Acad. Sci. Fennicae, xer. AI,2, 553–564 (1976).

Download references

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AS SSSR, Vol. 113, pp. 178–198, 1981.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tolokonnikov, V.A. Estimates in the Carleson corona theorem, ideals of the algebra H, a problem of S.-Nagy. J Math Sci 22, 1814–1828 (1983). https://doi.org/10.1007/BF01882580

Download citation


  • Hilbert Space
  • Linear Operator
  • Bounded Linear Operator
  • Minimal Norm
  • Inverse Element