# 2-summing multiplication operators

Studia Mathematica (2013)

• Volume: 216, Issue: 1, page 77-96
• ISSN: 0039-3223

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## Abstract

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Let 1 ≤ p < ∞, $={\left(Xₙ\right)}_{n\in ℕ}$ be a sequence of Banach spaces and ${l}_{p}\left(\right)$ the coresponding vector valued sequence space. Let $={\left(Xₙ\right)}_{n\in ℕ}$, $={\left(Yₙ\right)}_{n\in ℕ}$ be two sequences of Banach spaces, $={\left(Vₙ\right)}_{n\in ℕ}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator ${M}_{}:{l}_{p}\left(\right)\to {l}_{q}\left(\right)$ by ${M}_{}\left({\left(xₙ\right)}_{n\in ℕ}\right):={\left(Vₙ\left(xₙ\right)\right)}_{n\in ℕ}$. We give necessary and sufficient conditions for ${M}_{}$ 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 < ∞.

## How to cite

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Dumitru Popa. "2-summing multiplication operators." Studia Mathematica 216.1 (2013): 77-96. <http://eudml.org/doc/285383>.

@article{DumitruPopa2013,
abstract = {Let 1 ≤ p < ∞, $= (Xₙ)_\{n∈ℕ\}$ be a sequence of Banach spaces and $l_\{p\}()$ the coresponding vector valued sequence space. Let $= (Xₙ)_\{n∈ℕ\}$, $= (Yₙ)_\{n∈ℕ\}$ be two sequences of Banach spaces, $= (Vₙ)_\{n∈ℕ\}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_\{\}: l_\{p\}() → l_\{q\}()$ by $M_\{\}((xₙ)_\{n∈ℕ\}) : = (Vₙ(xₙ))_\{n∈ℕ\}$. We give necessary and sufficient conditions for $M_\{\}$ 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 < ∞.},
author = {Dumitru Popa},
journal = {Studia Mathematica},
keywords = {-summing; nuclear operators; operator ideals},
language = {eng},
number = {1},
pages = {77-96},
title = {2-summing multiplication operators},
url = {http://eudml.org/doc/285383},
volume = {216},
year = {2013},
}

TY - JOUR
AU - Dumitru Popa
TI - 2-summing multiplication operators
JO - Studia Mathematica
PY - 2013
VL - 216
IS - 1
SP - 77
EP - 96
AB - Let 1 ≤ p < ∞, $= (Xₙ)_{n∈ℕ}$ be a sequence of Banach spaces and $l_{p}()$ the coresponding vector valued sequence space. Let $= (Xₙ)_{n∈ℕ}$, $= (Yₙ)_{n∈ℕ}$ be two sequences of Banach spaces, $= (Vₙ)_{n∈ℕ}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{}: l_{p}() → l_{q}()$ by $M_{}((xₙ)_{n∈ℕ}) : = (Vₙ(xₙ))_{n∈ℕ}$. We give necessary and sufficient conditions for $M_{}$ 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 < ∞.
LA - eng
KW - -summing; nuclear operators; operator ideals
UR - http://eudml.org/doc/285383
ER -

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