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

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Abstract

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.

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Additional information

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

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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

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Keywords

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