J Austral Math Soc Ser A 45 pp401--420, 1988.

Scalar Operators and Integration

Igor Kluvánek

Abstract

The notion of a scalar operator on a Banach space, in the sense of N. Dunford, is widened so as to cover operators which can be approximated in the operator norm by linear combinations of disjoint values of an additive and multiplicative operator valued set function, P, on an algebra of sets in a space W such that P(W) = I, subject to some conditions guaranteeing that this definition is unambiguous. An operator T turns out to be scalar in this sense, if and only if, there exists a (not necessarily bounded) Boolean algebra of bounded projections such that the Banach algebra of operators it generates is semisimple and contains T.

1980 AMS Subject Classification (1985 Revision): 46G10, 47B40, 47D10, 43A22

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Authors

Igor Kluvánek

Centre for Mathematical Analysis, Australian National University, P. O. Box 4, Canberra, ACT 2600, Australia.