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p-VARIATION OF VECTOR MEASURES WITH RESPECT TO BILINEAR MAPS

Part of: Linear spaces and algebras of operators Measures, integration, derivative, holomorphy Special classes of linear operators Classical measure theory

Published online by Cambridge University Press:  01 December 2008

O. BLASCO  and
J. M. CALABUIG
O. BLASCO
Affiliation:
Department of Mathematics, Universitat de València, Burjassot 46100 (València), Spain (email: [email protected])
J. M. CALABUIG
Affiliation:
Department of Applied Mathematics, Universitat Politècnica de València, València 46022, Spain (email: [email protected])
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Abstract

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We introduce the spaces Vℬp(X) (respectively 𝒱ℬp(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×Y →Z and show that the spaces Lℬp(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 Vℬp(X). We characterize 𝒱ℬp(X) in terms of bilinear maps from Lp′×Y into Z and Vℬp(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.

Keywords

bilinear mapsvector integrationvector measuresumming operators

MSC classification

Secondary: 28A25: Integration with respect to measures and other set functions 46G10: Vector-valued measures and integration 47L05: Linear spaces of operators 47B10: Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 3 , December 2008 , pp. 411 - 430
Copyright
Copyright © 2009 Australian Mathematical Society

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