This is an announcement for the paper "Strong extensions for $q$-summing operators acting in $p$-convex function spaces for $1 \le p \le q$" by O. Delgado and E. A. Sanchez Perez.

Abstract: Let $1\le p\le q<\infty$ and let $X$ be a $p$-convex Banach function space over a $\sigma$-finite measure $\mu$. We combine the structure of the spaces $L^p(\mu)$ and $L^q(\xi)$ for constructing the new space $S_{X_p}^{,q}(\xi)$, where $\xi$ 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 $S_{X_p}^{,q}(\xi)$. This result turns out to be a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provide (strong) factorizations for $q$-summing operators through $L^q$-spaces when $1 \le q \le p$. Thus, our result completes the picture, showing what happens in the complementary case $1\le p\le q$, opening the door to the study of the multilinear versions of $q$-summing operators also in these cases.

Archive classification: math.FA

Mathematics Subject Classification: 46E30, 47B38

Submitted from: easancpe@mat.upv.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1506.09010

or

http://arXiv.org/abs/1506.09010