Second derivatives of norms and contractive complementation in vector-valued spaces

Bas Lemmens; Beata Randrianantoanina; Onno van Gaans

Displaying similar documents to “Second derivatives of norms and contractive complementation in vector-valued 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.

Structure of Rademacher subspaces in Cesàro type spaces

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 $C e s_{\infty}$ is investigated. It is shown that every infinite-dimensional subspace of either is isomorphic to l₂ and uncomplemented in $C e s_{\infty}$ , or contains a subspace isomorphic to c₀ and complemented in . The situation is rather different in the p-convexification of $C e s_{\infty}$ if 1 < p < ∞.

Three-space problems and bounded approximation properties

Wolfgang Lusky (2003)

Studia Mathematica

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Let ${R ₙ}_{n = 1}^{\infty}$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an $ℒ_{p}$ -space, then both X and A have bases. We apply these results to show that the spaces $C_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset C ()$ and $L_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset L ₁ ()$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.

The universal Banach space with a $K$ -suppression unconditional basis

Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2018)

Commentationes Mathematicae Universitatis Carolinae

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Using the technique of Fraïssé theory, for every constant $K \geq 1$ , we construct a universal object $𝕌_{K}$ in the class of Banach spaces possessing a normalized $K$ -suppression unconditional Schauder basis.

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

Elói Medina Galego (2004)

Colloquium Mathematicae

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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.

$L^{2}$ -Summand Vectors and Complemented Hilbertizable Subspaces

Antonio Aizpuru, Francisco J. García-Pacheco (2007)

Bollettino dell'Unione Matematica Italiana

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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.

Corrigendum to the paper “The universal Banach space with a $K$ -suppression unconditional basis”

Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2020)

Commentationes Mathematicae Universitatis Carolinae

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We observe that the notion of an almost ${𝔉ℑ}_{K}$ -universal based Banach space, introduced in our earlier paper [1]: Banakh T., Garbulińska-Wegrzyn J., The universal Banach space with a $K$ -suppression unconditional basis, Comment. Math. Univ. Carolin. 59 (2018), no. 2, 195–206, is vacuous for $K = 1$ . Taking into account this discovery, we reformulate Theorem 5.2 from [1] in order to guarantee that the main results of [1] remain valid.

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

Pilar Cembranos, Jose Mendoza (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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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.

Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K

Dale E. Alspach, Elói Medina Galego (2011)

Studia Mathematica

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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...

Addendum to "Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces" (Colloq. Math. 127 (2012), 105-109)

Aydin Sh. Shukurov (2014)

Colloquium Mathematicae

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It is well known that if φ(t) ≡ t, then the system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is a basis in some Lebesgue space $L_{p}$ . The aim of this short note is to show that the answer to this question is negative.

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.

Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces

Aydin Sh. Shukurov (2012)

Colloquium Mathematicae

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A necessary condition for Kostyuchenko type systems and system of powers to be a basis in $L_{p}$ (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in $L_{p}$ (1 ≤ p < +∞).

Rosenthal operator spaces

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

Studia Mathematica

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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...

On the number of non-isomorphic subspaces of a Banach space

Valentin Ferenczi, Christian Rosendal (2005)

Studia Mathematica

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We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let be a Banach space with an unconditional basis ${(e_{i})}_{i \in ℕ}$ ; then either there exists a perfect set P of infinite subsets of ℕ such that for any two distinct A,B ∈ P, ${[e_{i}]}_{i \in A} ≇ {[e_{i}]}_{i \in B}$ , or for a residual set of infinite subsets A of ℕ, ${[e_{i}]}_{i \in A}$ is isomorphic to , and in that case, is isomorphic to its square, to its hyperplanes, uniformly isomorphic to $\oplus {[e_{i}]}_{i \in D}$ for any D ⊂ ℕ, and isomorphic to a denumerable Schauder...

Uniqueness of unconditional basis of $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ , 0 < p < 1

F. Albiac, C. Leránoz (2002)

Studia Mathematica

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We prove that the quasi-Banach spaces $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ (0 < p < 1) have a unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss and Tzafriri have previously proved that the same is true for the respective Banach envelopes $ℓ ₁ (c ₀)$ and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

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

Artur Michalak (2003)

Studia Mathematica

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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...

On complemented copies of c₀(ω₁) in C(Kⁿ) spaces

Leandro Candido, Piotr Koszmider (2016)

Studia Mathematica

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Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $\otimes ̂_{ε}^{n} C (K)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $\otimes ̂_{ε}^{n} C (K)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X \otimes ̂_{ε} Y$ contains a complemented copy of c₀ if one of the infinite-dimensional...

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 $C e s_{p} [0, 1]$ .

On Lindenstrauss-Pełczyński spaces

Jesús M. F. Castillo, Yolanda Moreno, Jesús Suárez (2006)

Studia Mathematica

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We consider some stability aspects of the classical problem of extension of C(K)-valued operators. We introduce the class ℒ of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of c₀ into them can be extended to c₀. We show that all ℒ-spaces are of type $ℒ_{\infty}$ but not conversely. Moreover, $ℒ_{\infty}$ -spaces will be characterized as those spaces E such that E-valued operators from w*(l₁,c₀)-closed subspaces of l₁ extend to l₁. Regarding examples we will...

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

Dale E. Alspach, Simei Tong (2003)

Studia Mathematica

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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...

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)...

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...

A basis of Zₘ

Min Tang, Yong-Gao Chen (2006)

Colloquium Mathematicae

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Let $σ_{A} (n) = | (a, a^{'}) \in A ² : a + a^{'} = n |$ , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, $σ_{A} (n)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which $σ_{A} (n)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and $σ_{A} (n ̅) \leq 768$ for all n̅ ∈ Zₘ.

Decompositions for real Banach spaces with small spaces of operators

Manuel González, José M. Herrera (2007)

Studia Mathematica

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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...

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

Yasushi Hirata, Nobuyuki Kemoto (2003)

Fundamenta Mathematicae

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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.

Every separable Banach space has a basis with uniformly controlled permutations

Paolo Terenzi

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There exists a universal control sequence ${p ̅ (m)}_{m = 1}^{\infty}$ of increasing positive integers such that: Every infinite-dimensional separable Banach space X has a biorthogonal system xₙ,xₙ* with ||xₙ|| = 1 and ||xₙ*|| < K for each n such that, for each x ∈ X, $x = \sum_{n = 1}^{\infty} x_{π (n)} * (x) x_{π (n)}$ where π(n) is a permutation of n which depends on x but is uniformly controlled by ${p ̅ (m)}_{m = 1}^{\infty}$ , that is, $n_{n = 1}^{m} \subseteq π {(n)}_{n = 1}^{p ̅ (m)} \subseteq n_{n = 1}^{p ̅ (m + 1)}$ for each m.

The classical subspaces of the projective tensor products of $ℓ_{p}$ and C(α) spaces, α < ω₁

Elói Medina Galego, Christian Samuel (2013)

Studia Mathematica

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We completely determine the $ℓ_{q}$ and C(K) spaces which are isomorphic to a subspace of $ℓ_{p} \otimes ̂_{π} C (α)$ , the projective tensor product of the classical $ℓ_{p}$ space, 1 ≤ p < ∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α < ω₁. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from $ℓ_{p}$ to ℓ₁, 1 ≤ p < ∞. The first main theorem is an extension of a result of E. Oja and states...

On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems

Robert E. Zink (2002)

Colloquium Mathematicae

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In one of the earliest monographs that involve the notion of a Schauder basis, Franklin showed that the Gram-Schmidt orthonormalization of a certain Schauder basis for the Banach space of functions continuous on [0,1] is again a Schauder basis for that space. Subsequently, Ciesielski observed that the Gram-Schmidt orthonormalization of any Schauder system is a Schauder basis not only for C[0,1], but also for each of the spaces $L^{p} [0, 1]$ , 1 ≤ p < ∞. Although perhaps not probable, the latter...

$ℓ_{\infty}$ -sums and the Banach space $ℓ_{\infty} / c ₀$

Christina Brech, Piotr Koszmider (2014)

Fundamenta Mathematicae

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This paper is concerned with the isomorphic structure of the Banach space $ℓ_{\infty} / c ₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that $ℓ_{\infty} / c ₀$ does not have an orthogonal $ℓ_{\infty}$ -decomposition, that is, it is not of the form $ℓ_{\infty} (X)$ for any Banach space X. The main local result is that it is consistent that $ℓ_{\infty} (c ₀ ())$ does not embed isomorphically into $ℓ_{\infty} / c ₀$ , where is the cardinality of the continuum,...

Distortion and spreading models in modified mixed Tsirelson spaces

S. A. Argyros, I. Deliyanni, A. Manoussakis (2003)

Studia Mathematica

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The results of the first part concern the existence of higher order ℓ₁ spreading models in asymptotic ℓ₁ Banach spaces. We sketch the proof of the fact that the mixed Tsirelson space T[(ₙ,θₙ)ₙ], $θ_{n + m} \geq θ ₙ θ ₘ$ and $l i m_{n} θ ₙ^{1 / n} = 1$ , admits an $ℓ ₁^{ω}$ spreading model in every block subspace. We also prove that if X is a Banach space with a basis, with the property that there exists a sequence (θₙ)ₙ ⊂ (0,1) with $l i m_{n} θ ₙ^{1 / n} = 1$ , such that, for every n ∈ ℕ, $| | \sum_{k = 1}^{m} x_{k} | | \geq θ ₙ \sum_{k = 1}^{m} | | x_{k} | |$ for every ₙ-admissible block sequence ${(x_{k})}_{k = 1}^{m}$ of vectors in X, then there exists c...

General Haar systems and greedy approximation

Anna Kamont (2001)

Studia Mathematica

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We show that each general Haar system is permutatively equivalent in $L^{p} ([0, 1])$ , 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^{p} ([0, 1])$ , 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^{p} ({[0, 1]}^{d})$ , 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases...

Three-space problems for the approximation property

A. Szankowski (2009)

Journal of the European Mathematical Society

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It is shown that there is a subspace $Z_{q}$ of $ℓ_{q}$ for $1 < q < 2$ which is isomorphic to $ℓ_{q}$ such that $ℓ_{q} / Z_{q}$ does not have the approximation property. On the other hand, for $2 < p < \infty$ there is a subspace $Y_{p}$ of $ℓ_{p}$ such that $Y_{p}$ does not have the approximation property (AP) but the quotient space $ℓ_{p} / Y_{p}$ is isomorphic to $ℓ_{p}$ . The result is obtained by defining random “Enflo-Davie spaces” $Y_{p}$ which with full probability fail AP for all $2 < p \leq \infty$ and have AP for all $1 \leq p \leq 2$ . For $1 < p \leq 2$ , $Y_{p}$ are isomorphic to $ℓ_{p}$ .

Structure properties of D-R spaces

Hartmut von Trotha

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CONTENTSIntroduction................................................................................................................................... 5 Notations.......................................................................................................................... 5§ 1. Preliminaries........................................................................................................................ 6 1. Right invertible operators.....................................................................................................

A note on Dunford-Pettis like properties and complemented spaces of operators

Ioana Ghenciu (2018)

Commentationes Mathematicae Universitatis Carolinae

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Equivalent formulations of the Dunford-Pettis property of order $p$ ( ${D P P}_{p}$ ), $1 < p < \infty$ , are studied. Let $L (X, Y)$ , $W (X, Y)$ , $K (X, Y)$ , $U (X, Y)$ , and $C_{p} (X, Y)$ denote respectively the sets of all bounded linear, weakly compact, compact, unconditionally converging, and $p$ -convergent operators from $X$ to $Y$ . Classical results of Kalton are used to study the complementability of the spaces $W (X, Y)$ and $K (X, Y)$ in the space $C_{p} (X, Y)$ , and of $C_{p} (X, Y)$ in $U (X, Y)$ and $L (X, Y)$ .

Metric unconditionality and Fourier analysis

Stefan Neuwirth (1998)

Studia Mathematica

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We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces $L_{E}^{p} ()$ and $C_{E} ()$ of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces $p_{E} ()$ , p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L_{E}^{p} ()$ ...

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ} (X)$ ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ} (Y)$ ), and $ϕ_{\infty}$ a linear isometry from M into C(Y) (resp., $C_{ℝ} (Y)$ ). We show, under the assumption that $Π_{N} \subset Π_{T}$ , where $Π_{N}$ is...

A new function space and applications

Jean Bourgain, Haïm Brezis, Petru Mironescu (2015)

Journal of the European Mathematical Society

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We define a new function space $B$ , which contains in particular BMO, BV, and $W^{1 / p, p}$ , $1 < p < \infty$ . We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^{p}$ norms of integer-valued functions in $B$ . We introduce a significant closed subspace, $B_{0}$ , of $B$ , containing in particular VMO and $W^{1 / p, p}$ , $1 \leq p < \infty$ . The above mentioned estimates imply in particular that integer-valued functions belonging to $B_{0}$ are necessarily constant. This framework provides a “common roof”...

Operators on the stopping time space

Dimitris Apatsidis (2015)

Studia Mathematica

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Let S¹ be the stopping time space and ℬ₁(S¹) be the Baire-1 elements of the second dual of S¹. To each element x** in ℬ₁(S¹) we associate a positive Borel measure $μ_{x * *}$ on the Cantor set. We use the measures $μ_{x * *} : x * * \in ℬ ₁ (S ¹)$ to characterize the operators T: X → S¹, defined on a space X with an unconditional basis, which preserve a copy of S¹. In particular, if X = S¹, we show that T preserves a copy of S¹ if and only if $μ_{T * * (x * *)} : x * * \in ℬ ₁ (S ¹)$ is non-separable as a subset of $ℳ (2^{ℕ})$ .