Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T03:09:06.960Z Has data issue: false hasContentIssue false
  • English
  • Français

p-VARIATION OF VECTOR MEASURES WITH RESPECT TO BILINEAR MAPS

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

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])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Bartle, R. G., ‘A general bilinear vector integral’, Studia Math. 15 (1956), 337–352.CrossRefGoogle Scholar
[2]Blasco, O., ‘Positive p-summing operators, vector measures and tensor products’, Proc. Edinburgh Math. Soc. (2) 31(2) (1988), 179–184.Google Scholar
[3]Blasco, O., ‘Remarks on the semivariation of vector measures with respect to Banach spaces’, Bull. Austral. Math. Soc. 75(3) (2007), 469–480.CrossRefGoogle Scholar
[4]Blasco, O. and Calabuig, J. M., ‘Vector-valued functions integrable with respect to bilinear maps’, Taiwanese Math. J. to appear.Google Scholar
[5]Blasco, O. and Calabuig, J. M., ‘Fourier analysis with respect to bilinear maps’, Acta Math. Sinica to appear.Google Scholar
[6]Blasco, O. and Calabuig, J. M., ‘Hölder inequality for functions integrable with respect to bilinear maps’, Math. Scan. 102 (2008), 101–110.Google Scholar
[7]Diestel, J., Jarchow, H. and Tonge, A., Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, 43 (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
[8]Diestel, J. and Uhl Jr, J. J., Vector Measures, Mathematical Surveys, 15 (American Mathematical Society, Providence, RI, 1977), xiii+322 pp.CrossRefGoogle Scholar
[9]Dinculeanu, N., Vector Measures, International Series of Monographs in Pure and Applied Mathematics, 95 (Pergamon Press, Oxford/VEB Deutscher Verlag der Wissenschaften, Berlin, 1967).CrossRefGoogle Scholar
[10]Girardi, M. and Weis, L., ‘Integral operators with operator-valued kernels.’, J. Math. Anal. Appl. 290(1) (2004), 190–212.Google Scholar
[11]Jefferies, B. and Okada, S., ‘Bilinear integration in tensor products’, Rocky Mountain J. Math. 28(2) (1998), 517–545.Google Scholar
[12]Phillips, R. S., ‘On linear transformations’, Trans. Amer. Math. Soc. 48(290) (1940), 516–541.CrossRefGoogle Scholar
[13]Ryan, R. A., Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics (Springer, London, 2002).CrossRefGoogle Scholar
[14]Schaefer, H. H., Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, 215 (Springer, Heidelberg, 1974).CrossRefGoogle Scholar