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Multiple summing operators on $l_{p}$ spaces

100%
Popa D.
Studia Mathematica
|
2014
|
tom 225
|
nr 1
9-28
EN
We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_{p}$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_{p}$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T: l_{4/3} × l_{4/3} → l₂$ such that none of the associated linear operators is s-summing (1 ≤ s ≤ 2). Further we show that if n ≥ 2, there exist natural bounded multilinear operators $T: l_{2n/(n+1)} × ⋯ × l_{2n/(n+1)} → l₂$ for which none of the associated multilinear operators is multiple s-summing (1 ≤ s ≤ 2).
2
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Composition results for strongly summing and dominated multilinear operators

100%
Popa D.
Open Mathematics
|
2014
|
tom 12
|
nr 10
1433-1446
EN
In this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible n-linear versions of Grothendieck’s composition theorem in the case n ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact.
3
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2-summing multiplication operators

100%
Popa D.
Studia Mathematica
|
2013
|
tom 216
|
nr 1
77-96
EN
Let 1 ≤ p < ∞, $𝒳 = (Xₙ)_{n∈ℕ}$ be a sequence of Banach spaces and $l_{p}(𝒳)$ the coresponding vector valued sequence space. Let $𝒳 = (Xₙ)_{n∈ℕ}$, $𝓨 = (Yₙ)_{n∈ℕ}$ be two sequences of Banach spaces, $𝒱 = (Vₙ)_{n∈ℕ}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{𝒱}: l_{p}(𝒳) → l_{q}(𝓨)$ by $M_{𝒱}((xₙ)_{n∈ℕ}) : = (Vₙ(xₙ))_{n∈ℕ}$. We give necessary and sufficient conditions for $M_{𝒱}$ 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 < ∞.
4
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Rank α operators on the space C(T,X)

100%
Popa D.
Colloquium Mathematicum
|
2002
|
tom 91
|
nr 2
255-262
EN
For 0 ≤ α < 1, an operator U ∈ L(X,Y) is called a rank α operator if $xₙ → \limits^{τ_{α}} x$ implies Uxₙ → Ux in norm. We give some results on rank α operators, including an interpolation result and a characterization of rank α operators U: C(T,X) → Y in terms of their representing measures.
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