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Functional calculus, integral representations, and Banach space geometry. (English) Zbl 0815.47042

Summary: On suitable Banach spaces, the existence of a certain functional calculus for an operator is equivalent to that operator having a certain integral representation with respect to some family of projections. In this paper we consider the problem of identifying the precise classes of spaces on which these equivalences hold, for several of the most important functional calculi.

MSC:

47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47A60 Functional calculus for linear operators
46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
46J25 Representations of commutative topological algebras
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References:

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