Abstract. Every compact well-bounded operator has a representation as a linear combination of disjoint projections reminiscent of the representation of compact self-adjoint operators. In this note we show that the converse of this result holds, thus characterizing compact well-bounded operators. We also apply this result to study compact well-bounded operators on some special classes of Banach spaces such as hereditarily indecomposable spaces and certain spaces constructed by G. Pisier.1991 Mathematics Subject Classification. Primary 47B40. Secondary 34L10, 47A60, 47B07.1. Introduction. Well-bounded operators are defined as those which possess a functional calculus for the absolutely continuous functions on some compact interval ½a; b of the real line. Well-bounded operators were introduced by Smart [17] and Ringrose [14] in order to provide a theory for Banach space operators that was similar to the successful theory of self-adjoint operators on Hilbert space, but which included operators whose spectral expansions may only converge conditionally.On a general Banach space the integral representation theorems which one obtains for these operators are much less satisfactory than those obtained for selfadjoint operators. Nonetheless, even on an arbitrary Banach space, every compact well-bounded operator can be written in the form

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Well-bounded operators on general Banach spaces QINGPING CHENGThe theory of well-bounded operators has found many applications and has formed deep connections with other areas of mathematics. For example, it has been applied successfully to Sturm-Liouville theory, Fourier analysis and multiplier theory (see [1]). Although the theory of well-bounded operators is well established, there are a number of unresolved and interesting questions which are potentially fruitful areas for further research, and there are also a few errors in the literature. The general aim of this work is to answer some of these questions, to correct and clarify certain aspects of the theory, and to establish a more complete well-bounded operator theory, including a dual theory on general Banach spaces and a theory of compact well-bounded operators.We first give a general method to construct well-bounded operators with special properties, such as non-uniquely decomposable, non-decomposable in X and non type (B). This technique can be used to produce rich examples of well-bounded operators.An operator is well-bounded if and only if its adjoint is. On the other hand, if the underlying Banach space is nonreflexive, many of the porperties that a wellbounded operator may possess do not pass to the adjoint operator. One would hope that a suitable family of projections for the adjoint of a well-bounded operator could be formed by taking the adjoints of decomposition of the identity associated with this operator. We show that on a wide range of nonreflexive Banach spaces it is not possible to do this. We give a necessary and sufficient condition under which the hoped for relationship between corresponding decompositions of the identity holds. We have also included some results on quotients and restrictions of well-bounded operators. These results also appear in [4].In [9], Ringrose showed that if T is a well-bounded operator, there exists a decomposition of the identity for T whose projections commute with T*. He conjectured that it is possible for a well-bounded operator to have a decomposition of the identity whose projections do not all commute with T*. Turner [10] introduced scalar-type decomposable operators of class F and showed that this class of operators includes all adjoints

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 wellbounded 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.