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Composition operators on weighted spaces of continuous functions

Part of: Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

R. K. Singh  and
W. H. Summers
R. K. Singh
Affiliation:
Department of Mathematics, University of Jammu, Jammu-180 001, India
W. H. Summers
Affiliation:
Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701, U.S.A.
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Abstract

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We give algebraic criteria for distinguishing composition operators among all continuous linear operators on spaces of continuous functions with topologies generated by seminorms that are weighted analogues of the supremum norm. In another direction, we also characterize those self maps of the underlying topological space which induce composition operators on such weighted spaces, as well as determine conditions on these self maps which correspond to various basic properties of the induced composition operator. Our results are applied to a question concerning translation invariance which arises in the context of topological dynamics.

MSC classification

Secondary: 47B38: Operators on function spaces (general)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 3 , December 1988 , pp. 303 - 319
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Bierstedt, K.-D., Meise, R. G., and Summers, W. H., ‘A projective description of weighted inductive limits’, Trans. Amer. Math. Soc. 272 (1982), 107–160.CrossRefGoogle Scholar
[2]Gillman, L. and Jerison, M., Rings of continuous functions (Graduate Texts in Math. 43, Springer-Verlag, New York, 1976).Google Scholar
[3]Goullet de Rugy, A., ‘Espaces de fonctions pondérables’, Israel J. Math. 12 (1972), 147–160.CrossRefGoogle Scholar
[4]Hoover, T., Lambert, A., and Quinn, J., ‘The Markov process determined by a weighted composition operator’, Studia Math. 72 (1982), 225–235.CrossRefGoogle Scholar
[5]Kamowitz, H., ‘Compact weighted endomorphisms of C(X)’, Proc. Amer. Math. Soc. 83 (1981), 517–521.Google Scholar
[6]Koopman, B. O., ‘Hamiltonian systems and transformations in Hilbert space’, Proc. Nat. Acad. Sci. U.S.A. 17 (1931), 315–318.CrossRefGoogle ScholarPubMed
[7]Koopman, B. O. and Neumann, J. v., ‘Dynamical systems of continuous spectra’, Proc. Nat. Acad. Sci. U.S.A. 18 (1932), 255–263.CrossRefGoogle ScholarPubMed
[8]Mayer, D. H., ‘Spectral properties of certain composition operators arising in statistical mechanics’, Comm. Math. Phys. 68 (1979), 1–8.CrossRefGoogle Scholar
[9]Nachbin, L., Elements of approximation theory (Math. Studies 14, Van Nostrand, Princeton, N.J., 1967).Google Scholar
[10]Nordgren, E. A., ‘Composition operators on Hilbert spaces’, Hilbert space operators (Lecture Notes in Math., vol. 693, Springer-Verlag, Berlin, 1978, pp. 37–63).CrossRefGoogle Scholar
[11]Ruess, W. M. and Summers, W. H., ‘Minimal sets of almost periodic motions’, Math. Ann. 276 (1986), 145–158.CrossRefGoogle Scholar
[12]Schaefer, H. H., Topological vector spaces, (Graduate Texts in Math. 3, Springer-Verlag, New York, 1971).CrossRefGoogle Scholar
[13]Singh, R. K. and Summers, W. H., ‘Compact and weakly compact composition operators on spaces of vector valued continuous functions’, Proc. Amer. Math. Soc. 99 (1987), 667–670.CrossRefGoogle Scholar
[14]Summers, W. H., ‘A representation theorem for biequicontinuous completed tensor products of weighted spaces’, Trans. Amer. Math. Soc. 146 (1969), 121–131.CrossRefGoogle Scholar
[15]Webb, G. F., ‘Nonlinear semigroups and age-dependent population models’, Ann. Mat. Pura Appl. 129 (1981), 43–55.CrossRefGoogle Scholar