2-summing multiplication operators
Volume 216 / 2013
Abstract
Let $1\leq p<\infty $, $\mathcal {X}=(X_{n}) _{n\in \mathbb {N}}$ be a sequence of Banach spaces and $l_{p}(\mathcal {X}) $ the coresponding vector valued sequence space. Let $\mathcal {X}=( X_{n}) _{n\in \mathbb {N}}$, $\mathcal {Y}=(Y_{n}) _{n\in \mathbb {N}}$ be two sequences of Banach spaces, $\mathcal {V}=( V_{n}) _{n\in \mathbb {N}}$, $V_{n}:X_{n}\rightarrow Y_{n}$, a sequence of bounded linear operators and $1\leq p,q<\infty $. We define the multiplication operator $M_{\mathcal {V}}:l_{p}(\mathcal {X}) \rightarrow l_{q}(\mathcal {Y}) $ by $M_{\mathcal {V}}( (x_{n}) _{n\in \mathbb {N}}) :=(V_{n}( x_{n})) _{n\in \mathbb {N}}$. We give necessary and sufficient conditions for $M_{\mathcal {V}}$ to be $2$-summing when $(p,q) $ is one of the couples $(1,2) $, $(2,1) $, $(2,2) $, $( 1,1) $, $(p,1) $, $(p,2) $, $(2,p) $, $(1,p) $, $(p,q) $; in the last case $1< p< 2$, $1< q< \infty $.
