Geodesic
On Additive Operators
Canadian journal of mathematics, Tome 23 (1971) no. 3, pp. 468-480

Voir la notice de l'article provenant de la source Cambridge University Press

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh } which take their values in X**. If X is weakly sequentially complete, then {μh } can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]): 
Friedman, N. A.; Tong, A. E. On Additive Operators. Canadian journal of mathematics, Tome 23 (1971) no. 3, pp. 468-480. doi: 10.4153/CJM-1971-050-7
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