a1 Department of Mathematics, Universitat de València, Burjassot 46100 (València), Spain (email: [email protected])
a2 Department of Applied Mathematics, Universitat Politècnica de València, València 46022, Spain (email: [email protected])
We introduce the spaces Vp(X) (respectively p(X)) of the vector measures :Σ→X of bounded (p,)-variation (respectively of bounded (p,)-semivariation) with respect to a bounded bilinear map :X×Y →Z and show that the spaces Lp(X) consisting of functions which are p-integrable with respect to , 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 p(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,)-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
Keywords and phrases
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).