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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Well-bounded operators
on nonreflexive Banach spaces

Authors: Cheng Qingping and Ian Doust
Journal: Proc. Amer. Math. Soc. 124 (1996), 799-808
MSC (1991): Primary 47B40; Secondary 46B10, 46B15, 46B20, 47A60
MathSciNet review: 1301522
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Abstract: Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of well-bounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.

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Additional Information

Cheng Qingping
Affiliation: Department of Mathematics, Jingzhou Teachers College, Jingzhou, Hubei, People’s Republic of China

Ian Doust
Affiliation: School of Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia

Keywords: Well-bounded operators, functional calculus, nonreflexive Banach spaces
Received by editor(s): August 29, 1994
Additional Notes: This research was supported by the Australian Research Council.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society