Hostname: page-component-89b8bd64d-4ws75 Total loading time: 0 Render date: 2026-05-09T05:15:04.358Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-induced compactness in Banach spaces

Published online by Cambridge University Press:  14 November 2011

P. G. Casazza  and
H. Jarchow
P. G. Casazza
Affiliation:
Department of Mathematics, University of Missouri-Columbia, Columbia, Mo 65211, U.S.A. e-mail: Pete@casazza.cs.missouri.edu
H. Jarchow
Affiliation:
Mathematisches Institut, Universität Zürich, CH 8057 Zürich, Switzerland e-mail: jarchow@math.unizh.ch
Get access Rights & Permissions [Opens in a new window]

Extract

We consider the question: is every compact set in a Banach space X contained in the closed unit range of a compact (or even approximable) operator on X? We give large classes of spaces where the question has an affirmative answer, but observe that it has a negative answer, in general, for approximable operators. We further construct a Banach space failing the bounded compact approximation property, though all of its duals have the metric compact approximation property.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 2 , 1996 , pp. 355 - 362
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Bellenot, S.. The Schwartz Hilbert variety. Michigan Math. J. 22 (1975), 373–7.Google Scholar
2Bonsall, F. F. and Duncan, J.. Complete Nnrmed Algebras (Berlin: Springer. 1973).CrossRefGoogle Scholar
3Cho, C. and Johnson, W. B.. A characterization of subspaces X and lp for which K(X) is an M-ideai in L(X). Proc. Amer. Math. Soc. 93 (1985). 466–70.Google Scholar
4Cohen, P. J.. Factorization in group algebras. Duke Math. J. 26 (1959).199–205.CrossRefGoogle Scholar
5Dales, H. G. and Jarchow, H.. Continuity of homomorphisms and derivations from algebras ofapproximablc and nuclear operators. Math. Proc. Cambridge Philos. Soc. 116 (1994). 465–73.CrossRefGoogle Scholar
6Dixon, P. G.. Left approximate identities in algebras of compact operators on Banach spaces. Proc. Ray. Soc. Edinburgh Sea. A 104 (1986), 169–75.CrossRefGoogle Scholar
7Grønbæk, N. and Willis, G. A.. Approximate identities in the Banach algebras of compact operators. Canad. Math. Bull 36 (1993). 45–53.CrossRefGoogle Scholar
8Grothendieck, A.. Produits tensoriels topologies et espaces nucléates. Mem. Amer. Math. Soc. 16(1955).Google Scholar
9John, K.. On the compact non-nuclear problem. Math. Ann. 287 (1990). 509–14.CrossRefGoogle Scholar
10Johnson, W. B.. A complementary universal conjugate Banach space and its relation to the approximation problem. Israel J. Math. 13 (1972). 301–10.CrossRefGoogle Scholar
11Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces I, Sequence Spaces, Results in Mathematics and Related Areas 92 (Berlin: Springer, 1977).Google Scholar
12Morell, J. S. and Retherford, J. R.. p-trivial Banach spaces. Stadia Math. 43 (1972), 1–25.CrossRefGoogle Scholar
13Pietsch, A.. Operator Ideals (VEB Deutscher Verlag der Wissenschaftens 1978; Amsterdam:North-Holland, 1980).Google Scholar
14Pisier, G.. Un théorème sur les opérateurs entre espaces de Banach qui se factorisenl par un espace de Hilbert. Ann. Sci. École Norm- Sup. 13 (1980), 23–43.CrossRefGoogle Scholar
15Pisier, G.. Counterexample to a conjecture of Grothendicck. Acta Math. 151 (1983), 180–208.CrossRefGoogle Scholar
16Samuel, C-. Bounded approximate identities in the algebra of compact operators on a Banach space. Proc. Amer. Math. Soc. 117 (1993). 1093–6.CrossRefGoogle Scholar
17Selivanov, Y. V.. Homological characterizations of the approximation property for Banach spaces. Glasgow Math. J. 34 (1992), 229–39.CrossRefGoogle Scholar
18Szankowski, A.. Subspaces without the approximation property. Israel J. Math. 30 (1978), 123–9.CrossRefGoogle Scholar
19Willis, G. A.. The compact approximation property does not imply the approximation property. Studia Math. 103 (1992), 99–108.CrossRefGoogle Scholar