Well-bounded operators on nonreflexive Banach spaces

Qingping, Cheng; Doust, Ian

doi:10.1090/S0002-9939-96-03098-5

Well-bounded operators on nonreflexive Banach spaces
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by Cheng Qingping and Ian Doust PDF

Proc. Amer. Math. Soc. 124 (1996), 799-808 Request permission

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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