# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## 2-summing multiplication operators

### Tom 216 / 2013

Studia Mathematica 216 (2013), 77-96 MSC: Primary 47B10, 47L20; Secondary 46B45. DOI: 10.4064/sm216-1-6

#### Streszczenie

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 $1< p< 2$, $1< q< \infty$.