We present a Riesz type representation theorem for multilinear operators defined on the product of C(K,X) spaces with values in a Banach space. In order to do this we make a brief exposition of the theory of operator valued polymeasures.
We use polymeasures to characterize when a multilinear form defined on a product of spaces is integral.
Given Banach spaces , and a compact Hausdorff space , we use polymeasures to give necessary conditions for a multilinear operator from into to be completely continuous (resp. unconditionally converging). We deduce necessary and sufficient conditions for to have the Schur property (resp. to contain no copy of ), and for to be scattered. This extends results concerning linear operators.
The purpose of this note is to announce, without proofs, some results concerning vector valued multilinear operators on a product of C(K) spaces.
We obtain a classification of projective tensor products of C(K) spaces according to whether none, exactly one or more than one factor contains copies of ℓ₁, in terms of the behaviour of certain classes of multilinear operators on the product of the spaces or the verification of certain Banach space properties of the corresponding tensor product. The main tool is an improvement of some results of Emmanuele and Hensgen on the reciprocal Dunford-Pettis and Pełczyński's (V) properties of the projective...
In recent papers, the Right and the Strong* topologies have been introduced and studied on general Banach spaces. We characterize different types of continuity for multilinear operators (joint, uniform, etc.) with respect to the above topologies. We also study the relations between them. Finally, in the last section, we relate the joint Strong*-to-norm continuity of a multilinear operator T defined on C*-algebras (respectively, JB*-triples) to C*-summability (respectively, JB*-triple-summability)....
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