Bulletin of the Australian Mathematical Society

Research Article

p-VARIATION OF VECTOR MEASURES WITH RESPECT TO BILINEAR MAPS

O. BLASCOa1 and J. M. CALABUIGa2

a1 Department of Mathematics, Universitat de València, Burjassot 46100 (València), Spain (email: oscar.blasco@uv.es)

a2 Department of Applied Mathematics, Universitat Politècnica de València, València 46022, Spain (email: jmcalabu@mat.upv.es)

Abstract

We introduce the spaces Vxs212Cp(X) (respectively xs1D4B1xs212Cp(X)) of the vector measures xs2131:Σ→X of bounded (p,xs212C)-variation (respectively of bounded (p,xs212C)-semivariation) with respect to a bounded bilinear map xs212C:X×YZ and show that the spaces Lxs212Cp(X) consisting of functions which are p-integrable with respect to xs212C, defined in by Blasco and Calabuig [‘Vector-valued functions integrable with respect to bilinear maps’, Taiwanese Math. J. to appear], are isometrically embedded in Vxs212Cp(X). We characterize xs1D4B1xs212Cp(X) in terms of bilinear maps from Lp′×Y into Z and Vxs212Cp(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,xs212C)-variation of a measure in terms of the cone-absolutely summing norm of the corresponding bilinear map from Lp′×Y into Z.

(Received February 13 2008)

2000 Mathematics subject classification

  • 28A25;
  • 46G10;
  • 47L05;
  • 47B10

Keywords and phrases

  • bilinear maps;
  • vector integration;
  • vector measure;
  • summing operators

Footnotes

The authors gratefully acknowledge support from Spanish Grants MTM2005-08350-C03-03 and MTN2004-21420-E. J. M. Calabuig was also supported by Generalitat Valenciana (project GV/2007/191).