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Classes of operators on vector valued integration spaces

Part of: Linear function spaces and their duals Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

R. J. Fleming  and
J. E. Jamison
J. E. Jamison
Affiliation:
Department of Mathematics, Memphis State University Memphis Tennessee 38152 USA
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Abstract

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Let Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

MSC classification

Secondary: 46E40: Spaces of vector- and operator-valued functions 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 24 , Issue 2 , September 1977 , pp. 129 - 138
Copyright
Copyright © Australian Mathematical Society 1977

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