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Linear operators in l2 with a universality property

Part of: Maps and general types of spaces defined by maps Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

Michael Edelstein  and
Raymond D. Holmes
Michael Edelstein
Affiliation:
Dalhousie UnivesityHalifax, Nova Scotia, Canada
Raymond D. Holmes
Affiliation:
Dalhousie UnivesityHalifax, Nova Scotia, Canada
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Abstract

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A collection P of bounded linear operators in l2 is constructed in such a manner that given any separable metric space X, and any countable collection F of continuous self-maps of X, there is a homeomorphism h of X onto a subset of l2 such that for each f ∈ F there is P ∈ P with hf = Ph.

While similar results were obtained by Baayen and Dc Groot, our construction makes it possible to impose additional conditions on h (depending on F). For example, if all the members of F are uniformly continuous then h too can be made uniformly continuous.

MSC classification

Secondary: 46C05: Hilbert and pre-Hilbert spaces 54C05: Continuous maps
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 2 , April 1990 , pp. 214 - 222
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Baayen, P. C. and DeGroot, J., ‘Linearization of locally compact transformation groups in Hubert space,’ Math. Systems Theory 2 (1968), 363–379.CrossRefGoogle Scholar
[2]DeGroot, J. and McDowell, R. H., ‘Extension of mappings on metric spaces,’ Fund. Math. 48 (1960), 251–263.CrossRefGoogle Scholar
[3]Urysohn, P., ‘Zum Metrizations Problem,’ Math. Ann. 94 (1925), 309–315.CrossRefGoogle Scholar