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

Strong extensions for q-summing operators acting in p-convex Banach function spaces for $$1 \le... Let $$1\le p\le q<\infty $$ 1 ≤ p ≤ q < ∞ and let X be a p-convex Banach function space over a $$\sigma $$ σ -finite measure $$\mu $$ μ . We combine the structure of the spaces $$L^p(\mu )$$ L p ( μ ) and $$L^q(\xi )$$ L q ( ξ ) for constructing the new space $$S_{X_p}^{\,q}(\xi )$$ S X p q ( ξ ) , where $$\xi $$ ξ is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-summing operator T defined on X can be continuously (strongly) extended to $$S_{X_p}^{\,q}(\xi )$$ S X p q ( ξ ) . Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through $$L^q$$ L q -spaces when $$1 \le q \le p$$ 1 ≤ q ≤ p . Thus, our result completes the picture, showing what happens in the complementary case $$1\le p\le q$$ 1 ≤ p ≤ q . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

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Publisher
Springer International Publishing
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0397-1
Publisher site
See Article on Publisher Site

Abstract

Let $$1\le p\le q<\infty $$ 1 ≤ p ≤ q < ∞ and let X be a p-convex Banach function space over a $$\sigma $$ σ -finite measure $$\mu $$ μ . We combine the structure of the spaces $$L^p(\mu )$$ L p ( μ ) and $$L^q(\xi )$$ L q ( ξ ) for constructing the new space $$S_{X_p}^{\,q}(\xi )$$ S X p q ( ξ ) , where $$\xi $$ 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 $$S_{X_p}^{\,q}(\xi )$$ 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 $$L^q$$ L q -spaces when $$1 \le q \le p$$ 1 ≤ q ≤ p . Thus, our result completes the picture, showing what happens in the complementary case $$1\le p\le q$$ 1 ≤ p ≤ q .

Journal

PositivitySpringer Journals

Published: Feb 2, 2016

References

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