Abstract.
Let X, Y be Banach spaces. We say that a set \(\user2{\mathbb{M}}\, \subset \,\prod\nolimits_p {(X,\,Y)}\) is uniformly p–summing if the series \(\sum\limits_n {||\,T\,x_n \,||^p }\) is uniformly convergent for \(T\, \in \,\user2{\mathbb{M}}\) whenever (x n ) belongs to \(\ell _w^p \,(X)\). We consider uniformly summing sets of operators defined on a \(\user1{\mathcal{C}}\,(\Omega ,\,X)\)-space and prove, in case X does not contain a copy of c0, that \(\user2{\mathbb{M}}\) is uniformly summing iff \(\user2{\mathbb{M}}^\# \, = \,\{ T^\# :\user1{\mathcal{C}}(\Omega )\, \to \,\prod\nolimits_1 {(X,\,Y)\,:} \,T \in \user2{\mathbb{M}}\}\) is, where T (φ x) = (T#φ) x for all \(\varphi \, \in \,\user1{\mathcal{C}}(\Omega )\) and x∈X. We also characterize the sets \(\user2{\mathbb{M}}\) with the property that \(\user2{\mathbb{M}}^\#\) is uniformly summing viewed in \(\prod\nolimits_1 {(\user1{\mathcal{C}}(\Omega ),\,\user1{\mathcal{L}}(X,\,Y))}\).
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Received: 1 July 2005
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Delgado, J.M., Piñeiro, C. Uniformly summing sets of operators on spaces of vector–valued continuous functions. Arch. Math. 87, 141–153 (2006). https://doi.org/10.1007/s00013-005-1648-8
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DOI: https://doi.org/10.1007/s00013-005-1648-8