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 | References | Similar Articles | Additional Information

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

Email:
i.doust@unsw.edu.au

**Ian Doust**

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

Email:
cheng@prodigal.murdoch.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-96-03098-5

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