Well-bounded operators on nonreflexive Banach spaces

Qingping, C. and Doust, I. (1996) Well-bounded operators on nonreflexive Banach spaces. Proceedings of the American Mathematical Society, 124 (03). pp. 799-809.

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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.