Strong extensions for q-summing operators acting in p-convex Banach function spaces for 1≤p≤q

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 20; no. 4; pp. 999 - 1014
• Main Authors: Delgado, O., Pérez, E. A. Sánchez
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2016; Springer Nature B.V
• Subjects: Banach spaces; Calculus of Variations and Optimal Control; Optimization; Econometrics; Fourier Analysis; Mathematics; Mathematics and Statistics; Operator Theory; Potential Theory; Operator; Extension; 47B38; summing; 46E30; convex; 46B42; Factorization
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-016-0397-1

Let 1 ≤ p ≤ q < ∞ and let X be a p -convex Banach function space over a σ -finite measure μ . We combine the structure of the spaces L p ( μ ) and L q ( ξ ) for constructing the new space S X p q ( ξ ) , where ξ is a probability Radon measure on a certain compact set associated to X . We show some of its properties, and the relevant fact that every q -summing operator T defined on X can be continuously (strongly) extended to S X p q ( ξ ) . Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q -summing operators through L q -spaces when 1 ≤ q ≤ p . Thus, our result completes the picture, showing what happens in the complementary case 1 ≤ p ≤ q .