On weakly compact operators on $C_0(T)$

Panchapagesan, T.V. (1995) On weakly compact operators on $C_0(T)$. Accademia Peloritana dei Pericolanti, Classe di Scienze FF. MM. NN., LXXIII. pp. 41-56.

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Let $T$ be a locally compact Hausdorff space and let $C_0(T)={f:T \to \ \mathbb{C} \f$ is continuous and vanishes at infinity } be provided with the supremum norm.Let $X$ be a quasicomplete locally convex Hausdorff space. Suppose $u:C_0(T) \to \ X$ is a continuous linear operator. By refining the method adopted by Grothendick in [6] and by combining the integration technique of Bartle-Dunford-Schwartz [1], are obtained 32 characterizations for the operator $u$ to be weakly compact, several of which are new. The present method is so powerful as to deduce the isolated result of Dinculeanu and Kluvanek on the regular Borel extension of $\sigma \$ -additive locally convex space valued Baire measures as corollary of the main characterization theorem. Also is included an elegant proof of the range theorem of Tweddle on \sigma -additive vector measures.