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p-VARIATION OF VECTOR MEASURES WITH RESPECT TO BILINEAR MAPS

Published online by Cambridge University Press:  01 December 2008

O. BLASCO
Affiliation:
Department of Mathematics, Universitat de València, Burjassot 46100 (València), Spain (email: oscar.blasco@uv.es)
J. M. CALABUIG
Affiliation:
Department of Applied Mathematics, Universitat Politècnica de València, València 46022, Spain (email: jmcalabu@mat.upv.es)
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Abstract

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We introduce the spaces Vp(X) (respectively 𝒱p(X)) of the vector measures ℱ:Σ→X of bounded (p,ℬ)-variation (respectively of bounded (p,ℬ)-semivariation) with respect to a bounded bilinear map ℬ:X×YZ and show that the spaces Lp(X) consisting of functions which are p-integrable with respect to ℬ, defined in by Blasco and Calabuig [‘Vector-valued functions integrable with respect to bilinear maps’, Taiwanese Math. J. to appear], are isometrically embedded in Vp(X). We characterize 𝒱p(X) in terms of bilinear maps from Lp′×Y into Z and Vp(X) as a subspace of operators from Lp′(Z*) into Y*. Also we define the notion of cone absolutely summing bilinear maps in order to describe the (p,ℬ)-variation of a measure in terms of the cone-absolutely summing norm of the corresponding bilinear map from Lp′×Y into Z.

Type
Research Article
Copyright
Copyright © 2009 Australian Mathematical Society

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