Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. Fonf; Michael Lin; Przemysław Wojtaszczyk

Displaying similar documents to “Poisson's equation and characterizations of reflexivity of Banach spaces”

Prolongation of Poisson $2$ -form on Weil bundles

Norbert Mahoungou Moukala, Basile Guy Richard Bossoto (2016)

Archivum Mathematicum

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In this paper, $M$ denotes a smooth manifold of dimension $n$ , $A$ a Weil algebra and $M^{A}$ the associated Weil bundle. When $(M, ω_{M})$ is a Poisson manifold with $2$ -form $ω_{M}$ , we construct the $2$ -Poisson form $ω_{M^{A}}^{A}$ , prolongation on $M^{A}$ of the $2$ -Poisson form $ω_{M}$ . We give a necessary and sufficient condition for that $M^{A}$ be an $A$ -Poisson manifold.

Reflexivity and approximate fixed points

Eva Matoušková, Simeon Reich (2003)

Studia Mathematica

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A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_{k}}$ for which $s u p_{f \in S_{X *}} l i m s u p_{k \to \infty} f (x_{n_{k}})$ is attained at some f in the dual unit sphere $S_{X *}$ . A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f \in S_{X *}$ , there exists $g \in S_{X *}$ such that $l i m s u p_{n \to \infty} f (x ₙ) < l i m i n f_{n \to \infty} g (x ₙ)$ . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

Canonical Poisson-Nijenhuis structures on higher order tangent bundles

P. M. Kouotchop Wamba (2014)

Annales Polonici Mathematici

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Let M be a smooth manifold of dimension m>0, and denote by $S_{c a n}$ the canonical Nijenhuis tensor on TM. Let Π be a Poisson bivector on M and $Π^{T}$ the complete lift of Π on TM. In a previous paper, we have shown that $(T M, Π^{T}, S_{c a n})$ is a Poisson-Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from M to $T^{r} M$ have been studied and some properties were given. Furthermore, the canonical Nijenhuis tensors on $T^{A} M$ are described by A. Cabras and I. Kolář [Arch. Math. (Brno) 38 (2002),...

Sequentially Right Banach spaces of order $p$

Mahdi Dehghani, Mohammad B. Dehghani, Mohammad S. Moshtaghioun (2020)

Commentationes Mathematicae Universitatis Carolinae

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We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order $p$ , and those defined by the dual property, the sequentially Right $^{*}$ Banach spaces of order $p$ for $1 \leq p \leq \infty$ . These classes of Banach spaces are characterized by the notions of $L_{p}$ -limited sets in the corresponding dual space and $R_{p}^{*}$ subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space $X$ and a reflexive Banach...

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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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. A function $f : {[0, 1]}^{m} \to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f : {[0, 1]}^{m} \to X$ with respect to any norming subset there exists a separately increasing function $g : {[0, 1]}^{m} \to ℝ$ such that the sets of...

Involutivity of truncated microsupports

Masaki Kashiwara, Térésa Monteiro Fernandes, Pierre Schapira (2003)

Bulletin de la Société Mathématique de France

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Using a result of J.-M. Bony, we prove the weak involutivity of truncated microsupports. More precisely, given a sheaf $F$ on a real manifold and $k \in ℤ$ , if two functions vanish on ${SS}_{k} (F)$ , then so does their Poisson bracket.

Unicellularity of the multiplication operator on Banach spaces of formal power series

B. Yousefi (2001)

Studia Mathematica

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Let ${β (n)}_{n = 0}^{\infty}$ be a sequence of positive numbers and 1 ≤ p < ∞. We consider the space $ℓ^{p} (β)$ of all power series $f (z) = \sum_{n = 0}^{\infty} f ̂ (n) z ⁿ$ such that $\sum_{n = 0}^{\infty} {| f ̂ (n) |}^{p} {| β (n) |}^{p} < \infty$ . We give some sufficient conditions for the multiplication operator, $M_{z}$ , to be unicellular on the Banach space $ℓ^{p} (β)$ . This generalizes the main results obtained by Lu Fang [1].

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$ , belonging to the class $L^{p}(J,E)$ , $p>1$ . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

On the H-property and rotundity of Cesàro direct sums of Banach spaces

Saard Youyen, Suthep Suantai (2008)

Banach Center Publications

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In this paper, we define the direct sum ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ has the H-property if and only if each $X_{i}$ has the H-property, and ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ has the Schur property if and only if each $X_{i}$ has the Schur property. Moreover, we also show that ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ is rotund if and only if each $X_{i}$ is rotund.

Pisier's inequality revisited

Tuomas Hytönen, Assaf Naor (2013)

Studia Mathematica

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Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies $\int_{- 1, 1 ⁿ} | | \sum_{j = 1}^{n} \partial_{j} f_{j} {(ε) | |}^{p} d μ (ε) \leq^{p} \int_{- 1, 1 ⁿ} \int_{- 1, 1 ⁿ} | | \sum_{j = 1}^{n} δ_{j} Δ f_{j} (ε) {| |}^{p} d μ (ε) d μ (δ)$ , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${\partial_{j}}_{j = 1}^{n}$ and $Δ = \sum_{j = 1}^{n} \partial_{j}$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by $ⁿ_{p} (X)$ , we show that $ⁿ_{p} (X) \leq \sum_{k = 1}^{n} 1 / k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

One-parameter contractions of Lie-Poisson brackets

Oksana Yakimova (2014)

Journal of the European Mathematical Society

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We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $𝒜 = 𝕂 [𝔸^{n}]$ is said to be of Kostant type, if its centre $Z (𝒜)$ is freely generated by homogeneous polynomials $F_{1}, ..., F_{r}$ such that they give Kostant’s regularity criterion on $𝔸^{n} (d_{x} F_{i}$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$ ). If the initial Poisson algebra is of Kostant type and $F_{i}$ satisfy a certain degree-equality, then the contraction...

Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Pavel Etingof, Victor Ginzburg (2010)

Journal of the European Mathematical Society

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The hypersurface in $ℂ^{3}$ with an isolated quasi-homogeneous elliptic singularity of type ${\tilde{E}}_{r}, r = 6, 7, 8$ , has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type $E_{r}$ provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra $ℂ [x_{1}, x_{2}, x_{3}]$ to a noncommutative algebra with generators $x_{1}, x_{2}, x_{3}$ and the following 3 relations...

$ℓ_{\infty}$ -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,...

C*-algebras have a quantitative version of Pełczyński's property (V)

Hana Krulišová (2017)

Czechoslovak Mathematical Journal

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A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T : X \to Y$ is weakly compact. H. Pfitzner proved that $C^{*}$ -algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C (K)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover,...

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Eric Leichtnam, Xiang Tang, Alan Weinstein (2007)

Journal of the European Mathematical Society

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Let $X$ be a complex manifold with strongly pseudoconvex boundary $M$ . If $ψ$ is a defining function for $M$ , then $- log ψ$ is plurisubharmonic on a neighborhood of $M$ in $X$ , and the (real) 2-form $σ = i \partial \bar{\partial} (- log ψ)$ is a symplectic structure on the complement of $M$ in a neighborhood of $M$ in $X$ ; it blows up along $M$ . The Poisson structure obtained by inverting $σ$ extends smoothly across $M$ and determines a contact structure on $M$ which is the same as the one induced by the complex structure. When $M$ is compact, the Poisson structure...

Limited $p$ -converging operators and relation with some geometric properties of Banach spaces

Mohammad B. Dehghani, Seyed M. Moshtaghioun (2021)

Commentationes Mathematicae Universitatis Carolinae

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By using the concepts of limited $p$ -converging operators between two Banach spaces $X$ and $Y$ , $L_{p}$ -sets and $L_{p}$ -limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as $*$ -Dunford–Pettis property of order $p$ and Pelczyński’s property of order $p$ , $1 \leq p < \infty$ .

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 a modification of the Poisson integral operator

Dariusz Partyka (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Given a quasisymmetric automorphism $γ$ of the unit circle $𝕋$ we define and study a modification $P_{γ}$ of the classical Poisson integral operator in the case of the unit disk $𝔻$ . The modification is done by means of the generalized Fourier coefficients of $γ$ . For a Lebesgue’s integrable complexvalued function $f$ on $𝕋$ , $P_{γ} [f]$ is a complex-valued harmonic function in $𝔻$ and it coincides with the classical Poisson integral of $f$ provided $γ$ is the identity mapping on $𝕋$ . Our considerations are motivated by...

An extension of Mazur's theorem on Gateaux differentiability to the class of strongly α (·)-paraconvex functions

S. Rolewicz (2006)

Studia Mathematica

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Let (X,||·||) be a separable real Banach space. Let f be a real-valued strongly α(·)-paraconvex function defined on an open convex subset Ω ⊂ X, i.e. such that $f (t x + (1 - t) y) \leq t f (x) + (1 - t) f (y) + m i n [t, (1 - t)] α (| | x - y | |)$ . Then there is a dense $G_{δ}$ -set $A_{G} \subset Ω$ such that f is Gateaux differentiable at every point of $A_{G}$ .

On some properties of generalized Marcinkiewicz spaces

Evgeniy Pustylnik (2001)

Studia Mathematica

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We give a full solution of the following problems concerning the spaces $M_{φ} (X ⃗)$ : (i) to what extent two functions φ and ψ should be different in order to ensure that $M_{φ} (X ⃗) \neq M_{ψ} (X ⃗)$ for any nontrivial Banach couple X⃗; (ii) when an embedding $M_{φ} (X ⃗) ⊊ M_{ψ} (X ⃗)$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_{φ} (X ⃗)$ -space for some (unknown beforehand) Banach couple X⃗.

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair $(𝒳, 𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $Γ$ . The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p : 𝒜 \to 𝒳$ . The space $𝒜$ is equipped with a closed $2$ -form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

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...

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.

The weak Gelfand-Phillips property in spaces of compact operators

Ioana Ghenciu (2017)

Commentationes Mathematicae Universitatis Carolinae

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For Banach spaces $X$ and $Y$ , let $K_{w^{*}} (X^{*}, Y)$ denote the space of all $w^{*} - w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $w G P$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $w G P$ . We investigate whether the spaces $K_{w^{*}} (X^{*}, Y)$ and $X \otimes_{ϵ} Y$ have the $w G P$ property, when $X$ and $Y$ have the $w G P$ property.