Structure of Rademacher subspaces in Cesàro type spaces

Sergey V. Astashkin; Lech Maligranda

Displaying similar documents to “Structure of Rademacher subspaces in Cesàro type spaces”

Completely 1-complemented subspaces of the Schatten spaces $S^{p}$

Jean Roydor (2007)

Banach Center Publications

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We describe the subspaces of $S^{p}$ (1 ≤ p ≠ 2 < ∞) which are the range of a completely contractive projection.

The Daugavet property and translation-invariant subspaces

Simon Lücking (2014)

Studia Mathematica

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Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{Λ} (G)$ or $L ¹_{Λ} (G)$ and which quotients of the form $C (G) / C_{Λ} (G)$ or $L ¹ (G) / L ¹_{Λ} (G)$ have the Daugavet property. We show that $C_{Λ} (G)$ is a rich subspace of C(G) if and only if $Γ ∖ Λ^{- 1}$ is a semi-Riesz set. If $L ¹_{Λ} (G)$ is a rich subspace of L¹(G), then $C_{Λ} (G)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C (G) / C_{Λ} (G)$ has the Daugavet property if Λ is a Rosenthal set, and that $L ¹_{Λ} (G)$ is a poor subspace of L¹(G)...

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In this paper, we show a necessary and sufficient condition for a real Banach space to have an infinite dimensional subspace which is hilbertizable and complemented using techniques related to $L^{2}$ -summand vectors.

Structure of Cesàro function spaces: a survey

Sergey V. Astashkin, Lech Maligranda (2014)

Banach Center Publications

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Geometric structure of Cesàro function spaces $C e s_{p} (I)$ , where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that $C e s_{p} [0, 1]$ contains isomorphic and complemented copies of $l_{q}$ -spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces $C e s_{p} [0, 1]$ .

Rademacher functions in Cesàro type spaces

Sergei V. Astashkin, Lech Maligranda (2010)

Studia Mathematica

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The Rademacher sums are investigated in the Cesàro spaces $C e s_{p}$ (1 ≤ p ≤ ∞) and in the weighted Korenblyum-Kreĭn-Levin spaces $K_{p, w}$ on [0,1]. They span l₂ space in $C e s_{p}$ for any 1 ≤ p < ∞ and in $K_{p, w}$ if and only if the weight w is larger than $t l o g ₂^{p / 2} (2 / t)$ on (0,1). Moreover, the span of the Rademachers is not complemented in $C e s_{p}$ for any 1 ≤ p < ∞ or in $K_{1, w}$ for any quasi-concave weight w. In the case when p > 1 and when w is such that the span of the Rademacher functions is isomorphic to l₂, this span is...

Rosenthal operator spaces

M. Junge, N. J. Nielsen, T. Oikhberg (2008)

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In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an $L_{p}$ -space, then it is either an $L_{p}$ -space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non-Hilbertian complemented operator subspaces of non-commutative $L_{p}$ -spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < ∞ which...

Matrix subspaces of L₁

Gideon Schechtman (2013)

Studia Mathematica

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If $E = e_{i}$ and $F = f_{i}$ are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices $a_{i, j}$ with norm $| | a_{i, j} {| |}_{E (F)} = | | \sum_{k} | | \sum_{l} a_{k, l} f_{l} | | e_{k} | |$ embeds into L₁. This generalizes a recent result of Prochno and Schütt.

Separating by $G_{δ}$ -sets in finite powers of ω₁

Yasushi Hirata, Nobuyuki Kemoto (2003)

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It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint $G_{δ}$ -sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of ⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of ⟨α,β,γ⟩ ∈ ω₁³: α < β < γ have this property.

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Dale E. Alspach, Elói Medina Galego (2011)

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A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C (ω^{ω})$ then X contains a copy...

On pairs of Banach spaces which are isomorphic to complemented subspaces of each other

Elói Medina Galego (2004)

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We establish the existence of Banach spaces E and F isomorphic to complemented subspaces of each other but with $E^{m} \oplus F ⁿ$ isomorphic to $E^{p} \oplus F^{q}$ , m, n, p, q ∈ ℕ, if and only if m = p and n = q.

On the mutually non isomorphic $l_{p} (l_{q})$

Pilar Cembranos, Jose Mendoza (2016)

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In this note we survey the partial results needed to show the following general theorem: $l_{p} (l_{q}) : 1 \leq p, q \leq + \infty$ is a family of mutually non isomorphic Banach spaces. We also comment some related facts and open problems.

On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity

Artur Michalak (2003)

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\bar{l i n f ([0, 1])}$ contains an order isomorphic copy of D(0,1), (2) $\bar{l i n f (Q)}$ contains an isomorphic copy of C([0,1]), (3) $\bar{l i n f ([0, 1])} / \bar{l i n f (Q)}$ contains an isomorphic copy of c₀(Γ) for some uncountable...

Subspaces of $L_{p}$ , p > 2, determined by partitions and weights

Dale E. Alspach, Simei Tong (2003)

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Many of the known complemented subspaces of $L_{p}$ have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of $L_{p}$ . It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given...

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Manuel González, José M. Herrera (2007)

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We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_{i}$ for which $(X_{i}) / ℐ n (X_{i})$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_{i}$ can be divided into subsets in such a way that if $X_{i}$ and $X_{j}$ are in different...

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

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Janko Bračič, Lina Oliveira (2018)

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Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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Let $W$ be the subspace of $ℕ^{*}$ consisting of all weak $P$ -points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$ -pseudocompact space for all $p \in ℕ^{*}$ .

A criterion for pure unrectifiability of sets (via universal vector bundle)

Silvano Delladio (2011)

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Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let $π_{V}$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an $^{m}$ -measurable subset of ℝⁿ with $^{m} (A) < \infty$ . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V, v) | V \in G (n, m), v \in V$ such that, for all P ∈ A, one has $^{m (n - m)} (V \in G (n, m) | (V, π_{V} (P)) \in Z) > 0$ . One can replace “for all P ∈ A” by “for...