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Scalar operators and integration

Part of: Special classes of linear operators Measures, integration, derivative, holomorphy Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Igor Kluvánek
Igor Kluvánek
Affiliation:
Centre for Mathematical Analysis, Australian National UniversityP. O. Box 4, Canberra, ACT 2600, Australia
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Abstract

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The notion of a scalar operator on a Banach space, in the sense of N. Dunford, is widened so as to cover those 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 Ω such that P(Ω) = 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.

MSC classification

Secondary: 46G10: Vector-valued measures and integration 47B40: Spectral operators, decomposable operators, well-bounded operators, etc. 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 3 , December 1988 , pp. 401 - 420
Copyright
Copyright © Australian Mathematical Society 1988

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