Properties of representations of operators acting between spaces of vector-valued functions

Properties of representations of operators acting between spaces of vector-valued functions A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued L 1-spaces into L ∞-spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on L p -spaces of functions with values in separable Banach spaces are precisely the multiplication operators. We extend this result to non-separable dual spaces. Moreover, we relate positivity and other order properties of the operators to corresponding properties of the representations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Properties of representations of operators acting between spaces of vector-valued functions

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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2010 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Econometrics; Operator Theory; Calculus of Variations and Optimal Control; Optimization; Fourier Analysis; Potential Theory
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0045-0
Publisher site
See Article on Publisher Site

Abstract

A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued L 1-spaces into L ∞-spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on L p -spaces of functions with values in separable Banach spaces are precisely the multiplication operators. We extend this result to non-separable dual spaces. Moreover, we relate positivity and other order properties of the operators to corresponding properties of the representations.

Journal

PositivitySpringer Journals

Published: Feb 9, 2010

References

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