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Composition operators on a functional Hilbert space

Published online by Cambridge University Press:  17 April 2009

R.K. Singh  and
S.D. Sharma
R.K. Singh
Affiliation:
Department of Mathematics, University of Jammu, Jammu, Tawi, India.
S.D. Sharma
Affiliation:
Department of Mathematics, University of Jammu, Jammu, Tawi, India.
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Abstract

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Let T be a mapping from a set X into itself and let H(X) be a functional Hilbert space on the set X. Then the composition operator CT on H(X) induced by T is a bounded linear transformation from H(X) into itself defined by CTf = f ∘ T. In this paper composition operators are characterized in the case when H(X) = H2(π+) in terms of the behaviour of the inducing functions in the vicinity of the point at infinity. An estimate for the lower bound of ∥CT∥ is given. Also the invertibility of CT is characterized in terms of the invertibility of T.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 20 , Issue 3 , June 1979 , pp. 377 - 384
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Caughran, James G. and Schwartz, Howard J., “Spectra of compact composition operators”, Proc. Amer. Math. Soc. 51 (1975), 127–130.CrossRefGoogle Scholar
[2]Duren, Peter L., Theory of Hp spaces (Pure and Applied Mathematics, 38. Academic Press, New York and London, 1970).Google Scholar
[3]Halmos, Paul R., A Hilbert space problem book (Van Nostrand, Princeton, New Jersey; Toronto; London; 1967. Reprinted: Graduate Texts in Mathematics, 19. Springer-Verlag, New York, Heidelberg, Berlin, 1974).Google Scholar
[4]Hoffman, Kenneth, Banach spaces of analytic functions (Prentice-Hall, Englewood Cliffs, New Jersey, 1962).Google Scholar
[5]Nordgren, Eric A., “Composition operators on Hilbert spaces”, Hilbert space operators, 37–63 (Proc. Conf. Hilbert Space Operators, California State University, Long Beach, 1977. Lecture Notes in Mathematics, 693. Springer-Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar
[6]Schwartz, Howard J., “Composition operators on Hp” (PhD thesis, University of Toledo, Ohio, 1969).Google Scholar
[7]Shapiro, J.H. & Taylor, P.D., “Compact, nuclear, and Hilbert-Schmidt composition operators on H2”, Indiana Univ. Math. J. 23 (1973), 471–496.CrossRefGoogle Scholar
[8]Singh, R.K., “A relation between composition operators on H 2(D) and H 2(Π+)”, Pure and Appl. Math. Sci. 1 (1974/1975), no. 2, 1–5.Google Scholar
[9]Singh, R.K., “Inner functions and composition operators on a Hardy space”, preprint.Google Scholar