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¦G¦∞∶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
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.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
P. Koosis, “Introduction to Hp spaces,” London Math. Soc. Lect. Notes Ser. No. 40 (1980).
N. K. Nikol'skii, Lectures on the Shift Operator [in Russian], Nauka, Moscow (1980).
M. Rosenblum, “A corona theorem for countably many functions,” Integral Equations and Operator Theory,3, No. 1, 125–137 (1980).
B. S. Nagy, “A problem on operator valued bounded analytic functions,” Zap. Nauch. Sem. Leningr. Otd. Mat. Inst.,81, 99 (1978).
R. Bellman, Introduction to Matrix Theory [Russian translation], Nauka, Moscow (1969).
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).
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).
R. G. Douglas, “Banach algebra techniques in the theory of Toeplitz operators,” CBMS Regional Conference,15, American Math. Soc., Providence, R. I. (1973).
F. Forelli, “Bounded holomorphic functions and projections,” Ill. J. Math.,10, 367–380 (1966).
D. Sarason, “Function theory on the unit circle,” Lect. Notes, Virginia (1978).
R. Rao, “On a generalized corona problem,” J. Anal. Math.,18, 277–278 (1967).
C. F. Shubert, “The corona theorem as an operator theorem,” Proc. Am. Math. Soc.,69, No. 1, 73–76 (1978).
B. S. Nagy and C. Foias, “On contractions similar to isometries and Toeplitz operators,” Ann. Acad. Sci. Fennicae, xer. AI,2, 553–564 (1976).
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AS SSSR, Vol. 113, pp. 178–198, 1981.
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
- Hilbert Space
- Linear Operator
- Bounded Linear Operator
- Minimal Norm
- Inverse Element