### Operators into ${L}_{p}$ which factor through ${l}_{p}$

W. B. Johnson (1979-1980)

Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")

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W. B. Johnson (1979-1980)

Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")

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B. L. Chalmers, F. T. Metcalf (1992)

Annales Polonici Mathematici

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It follows easily from a result of Lindenstrauss that, for any real twodimensional subspace v of L¹, the relative projection constant λ(v;L¹) of v equals its (absolute) projection constant $\lambda \left(v\right)=su{p}_{X}\lambda (v;X)$. The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in ${L}^{\infty}\left(\nu \right)$ and isometric to v and a projection ${P}_{\infty}$ from C ⊕ V onto V such that $\parallel {P}_{\infty}\parallel =\parallel P\u2081\parallel $, where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if $P\u2081={\sum}_{i=1}^{2}{U}_{i}\otimes {v}_{i}$, then ${P}_{\infty}={\sum}_{i=1}^{2}{u}_{i}\otimes {V}_{i}$, where $d{V}_{i}=2{v}_{i}d\nu $ and $d{U}_{i}=-2{u}_{i}d\nu $.

Brian Jefferies, Susumu Okada (2005)

Commentationes Mathematicae Universitatis Carolinae

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Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X{\widehat{\otimes}}_{\tau}Y$ is their complete tensor product with respect to some tensor product topology $\tau $. A uniformly bounded $X$-valued function need not be integrable in $X{\widehat{\otimes}}_{\tau}Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau $ is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index $1\le p<\infty $ and suppose that $X$ and $Y$ are ${L}^{p}$-spaces with ${\tau}_{p}$ the associated...

R. Zaharopol (1984)

Annales scientifiques de l'Université de Clermont-Ferrand 2. Série Probabilités et applications

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Elói Medina Galego (2013)

Colloquium Mathematicae

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Suppose that X and Y are Banach spaces that embed complementably into each other. Are X and Y necessarily isomorphic? In this generality, the answer is no, as proved by W. T. Gowers in 1996. However, if X contains a complemented copy of its square X², then X is isomorphic to Y whenever there exists p ∈ ℕ such that ${X}^{p}$ can be decomposed into a direct sum of ${X}^{p-1}$ and Y. Motivated by this fact, we introduce the concept of (p,q,r) widely complemented subspaces in Banach spaces, where p,q and...

Bas Lemmens, Beata Randrianantoanina, Onno van Gaans (2007)

Studia Mathematica

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We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces ${\ell}_{p}\left(X\right)$, where X is a Banach space with a 1-unconditional basis and p ∈ (1,2) ∪ (2,∞). If the norm of X is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of ${\ell}_{p}\left(X\right)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection...

Juan Carlos Ferrando (2000)

Commentationes Mathematicae Universitatis Carolinae

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If $(\Omega ,\Sigma ,\mu )$ is a finite measure space and $X$ a Banach space, in this note we show that ${L}_{{w}^{*}}^{1}(\mu ,{X}^{*})$, the Banach space of all classes of weak* equivalent ${X}^{*}$-valued weak* measurable functions $f$ defined on $\Omega $ such that $\parallel f\left(\omega \right)\parallel \le g\left(\omega \right)$ a.e. for some $g\in {L}_{1}\left(\mu \right)$ equipped with its usual norm, contains a copy of ${c}_{0}$ if and only if ${X}^{*}$ contains a copy of ${c}_{0}$.

Sergey V. Astashkin, Lech Maligranda (2015)

Studia Mathematica

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The structure of the closed linear span of the Rademacher functions in the Cesàro space $Ce{s}_{\infty}$ is investigated. It is shown that every infinite-dimensional subspace of either is isomorphic to l₂ and uncomplemented in $Ce{s}_{\infty}$, or contains a subspace isomorphic to c₀ and complemented in . The situation is rather different in the p-convexification of $Ce{s}_{\infty}$ if 1 < p < ∞.