Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-20T15:15:07.701Z Has data issue: false hasContentIssue false
  • English
  • Français

Nest Algebras of Operators and the Dunford-Pettis Property

Published online by Cambridge University Press:  20 November 2018

Timothy G. Feeman
Timothy G. Feeman*
Affiliation:
Villanova University, Villanova, PA 19085 USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A Banach space X is said to have the Dunford-Pettis Property if every weakly compact linear operator T: X —> Y, where Y is any Banach space, is completely continuous (that is, T maps weakly convergent sequences to strongly convergent ones). In this paper, we prove that if A is a nest algebra of operators on a separable, infinite dimensional Hilbert space, then A fails to have the Dunford-Pettis Property. We also investigate a certain algebra associated to A, analogous to a construction used by Bourgain and others in connection with the Dunford-Pettis Property for function algebras. We show that this algebra must lie between A and the quasi-triangular algebra A + K and we give examples to show that either extreme or something in between is possible. Finally, we consider the algebra of analytic Toeplitz operators and give a result for the corresponding associated algebra which is analogous to a result of Cima, Jansen, and Yale for H∞.

Keywords

47D2546B20.
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 2 , 01 June 1991 , pp. 208 - 214
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. William Arveson, Interpolation problems in nest algebras, J. Func. Anal., 20 (1975), 208–233.Google Scholar
2. Brown, S., Chevreau, B., and Pearcy, C., Contractions with rich spectrum have invariant subspaces, J. Operator. Th., 1 (1979), 123–136.Google Scholar
3. Bourgain, J., The Dunford-Pettis property for the ball-algebras, the polydisc-algebras, and the Sobolev spaces, Studia Math., 77 (1984), 245–253.Google Scholar
4. Bourgain, J., New Banach space properties of the disc algebra and H∞ , Acta Math., 152 (1984), 1–48.Google Scholar
5. Cima, J. A. and Timoney, R. M., The Dunford-Pettis propertyfor certain planar uniform algebras, Michigan Math. J., 34 (1987), 99–104.Google Scholar
6. Cima, J. A., Janson, S., and Yale, K., Completely continuous Hankel operators on H∞ and Bourgain algebras, preprint.Google Scholar
7. Davidson, K. R., The distance to the analytic Toeplitz operators, Illinois J. Math., to appear.Google Scholar
8. Davidson, K. R., Similarity and compact perturbations of nest algebras, J. fur die Reine und Angew. Math., 348 (1984), 72–87.Google Scholar
9. Diestel, J., A survey of results related to the Dunford-P ettis property, in Integration, Topology, and Geometry in Linear Spaces, W. Graves, H., editor, Contemporary Mathematics series, volume 2, American Math. Soc, Providence, 1980.Google Scholar
10. Dunford, N. and Pettis, B. J., Linear operations on summable functions, Trans. Amer. Math. Soc, 47 (1940), 323–392.Google Scholar
11. Erdos, J. A., Operators of finite rank in nest algebras, J. London Math. Soc, 43 (1968), 391–397.Google Scholar
12. Cecelia Laurie and Longstaff, W. E., A note on rank-one operators in reflexive algebras, Proc Amer. Math. Soc, 89(1983), 293–297.Google Scholar
13. Schatten, Robert, Norm ideals of completely continuous operators, Ergebnisse der Math, und ihrer Grenzgebiete, band 27, Springer-Verlag, New York Heidelberg Berlin, 1960.Google Scholar