Narcisse Randrianantoanina
Let $E$ be a separable Banach space and $Ω$ be a compact Hausdorff space. It is shown that the space $C(Ω,E)$ has property (V) if and only if $E$ does. Similar result is also given for Bochner spaces $L^p(μ,E)$ if $1<p<\infty$ and $μ$ is a finite Borel measure on $Ω$.
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We show that if X is a tight subspace of C(K) then X has the Pelczynski property and X^* is weakly sequentially complete. We apply this result to the space U of uniformly convergent Taylor series on the unit circle and using a minimal amount of Fourier theory prove a theorem of Bourgain, namely that U has the Pelczynski property and U^* is weakly sequentially complete. Using separate methods, we prove U and U^* have the Dunford-Pettis property. Some results concerning pointwise bounded approximation are proved for tight uniform algebras. We use tightness and the Pelczynski property sto make a …
A Banach space X has Pelczynski's property (V) if for every Banach space Y every unconditionally converging operator T: X -> Y is weakly compact. H. Pfitzner proved that C*-algebras have Pelczynski's property (V). In the preprint "H. Krulisova: Quantification of Pelczynski's property (V)" 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, we prove that in dual Banach spaces a quantitative …
Let $X$ be a Banach space and $(f_n)_n$ be a bounded sequence in $L^1(X)$. We prove a complemented version of the celebrated Talagrand's dichotomy i.e we show that if $(e_n)_n$ denotes the unit vector basis of $c_0$, there exists a sequence $g_n \in \text{conv}(f_n,f_{n+1},\dots)$ such that for almost every $ω$, either the sequence $(g_n(ω) \otimes e_n)$ is weakly Cauchy in $X \widehat{\otimes}_πc_0$ or it is equivalent to the unit vector basis of $\ell^1$. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of $\ell^1$ in $L^1(X)$. As an a…
Let M be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec-Pelczynski subsequence decomposition of bounded sequences in L^p[0,1] to the case of the Haagerup L^p-spaces (1\le p<\infty). In particular, we prove that if (ϕ_n)_n is a bounded sequence in the predual M_* of M, then there exist a subsequence (ϕ_{n_k})_k of (ϕ_n)_n, a decomposition ϕ_{n_k}= y_k+ z_k such that {y_k, k\ge 1} is relatively weaklycompact and the support projections s(z_k)\downarrow_k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexi…
We prove that in the setting of operator spaces the result of Davis, Figiel, Johnson and Pelczynski on factoring weakly compact operators holds accordingly. Though not related directly to the main theorem we add a remark on the description of weakly compact subsets in the dual of noncommutative vector valued ${\rm L}_1$.
We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak$^*$ analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of $L_1[0,1]$ over an $\ell_1$-subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pelczynski in the negative.
Pełczyński's property (V$^*$) in Lipschitz-free spaces
本文证明了利普希茨自由空间中佩尔切尼茨属性(V$^*$)的局部决定性，并获得若干充分条件。证明当完全度量空间M局部紧致且纯1-不可测，或为希尔伯特空间，或属于满足双霍尔德条件的卡诺-卡拉泰多里空间类时，F(M)具有属性(V$^*$)。
A Banach space $X$ has Pełczy\' nski's property (V) if for every Banach space $Y$ every unconditionally converging operator $T\colon X\to Y$ is weakly compact. In 1962, Aleksander Pełczy\' nski showed that $C(K)$ spaces for a compact Hausdorff space $K$ enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about $C(K)$ spaces and …
Banach Spaces with the Lebesgue Property of Riemann Integrability
本文定义了 Banach 空间的 Lebesgue 性质，利用介于 spreading 和 asymptotic models 间的新序列渐近结构对其进行刻画，并重现了 Pelczynski 和 da Rocha Filho 关于 $L_1[0,1]$ 子空间该性质的遗失定理。
We present a generalization of the Radon-Riesz property to sequences of continuous functions with values in uniformly convex and uniformly smooth Banach spaces.
This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as a geometrical property of a Banach space which ensures the space to have the Mazur-Ulam property. We prove that if a Banach space $X$ enjoys this property if and only if $C(K,X)$ enjoys this property. We also show the same result holds for $L_\infty(μ,X)$ and $L_1(μ,X)$.
If $μ_1,μ_2,\dots$ are positive measures on a measurable space $(X,Σ)$ and $v_1,v_2, \dots$ are elements of a Banach space ${\mathbb E}$ such that $\sum_{n=1}^\infty \|v_n\| μ_n(X) < \infty$, then $ω(S)= \sum_{n=1}^\infty v_n μ_n(S)$ defines a vector measure of bounded variation on $(X,Σ)$. We show ${\mathbb E}$ has the Radon-Nikodym property if and only if every ${\mathbb E}$-valued measure of bounded variation on $(X,Σ)$ is of this form. As an application of this result we show that under natural conditions an operator defined on positive measures, has a unique extension to an operator defin…
Let $( X,d )$ be a metric space with at least two elements and $( A , p_l )$ be a commutative semisimple Frechet algebra over the scalar field of complex numbers. The correlation between the BSE-property of the Frechet algebra $( A , p_l )$ and $\Lip_{d}( X , A )$ is assessed. It is found and approved that if $\Lip_{d}( X , A )$ is a BSE-Frechet algebra, then so is $A$. The opposite correlation will hold if $( A , p_l )$ is unital.
We prove that if a metric space $M$ has the finite CEP then $\mathcal F(M)\widehat{\otimes}_π X$ has the Daugavet property for every non-zero Banach space $X$. This applies, for instance, if $M$ is a Banach space whose dual is isometrically an $L_1(μ)$ space. If $M$ has the CEP then $L(\mathcal F(M),X)=\Lip(M,X)$ has the Daugavet property for every non-zero Banach space $X$, showing that this is the case when $M$ is an injective Banach space or a convex subset of a Hilbert space.
Let $K$ be a compact Hausdorff space and let $H$ be a real or complex Hilbert space with dim$(H_\mathbb{R})\geq 2$. We prove that the space $C(K,H)$ of all $H$-valued continuous functions on $K$, equipped with the supremum norm, satisfies the Mazur--Ulam property, that is, if $Y$ is any real Banach space, every surjective isometry $Δ$ from the unit sphere of $C(K,H)$ onto the unit sphere of $Y$ admits a unique extension to a surjective real linear isometry from $C(K,H)$ onto $Y$. Our strategy relies on the structure of $C(K)$-module of $C(K,H)$ and several results in JB$^*$-triple theory. For …
Compact Bloch mappings on the complex unit disc
本文引入Bloch分子和无Bloch巴拿赫空间，定义紧Bloch映射并建立其性质，给出经典相关定理的Bloch版本，研究复单位圆盘上Bloch函数空间的对偶性。
Given a Banach space $X$, we consider Cesàro spaces $\text{Ces}_p(X)$ of $X$-valued functions over the interval $[0,1]$, where $1\leq p<\infty$. We prove that if $X$ has the Opial/uniform Opial property, then certain analogous properties also hold for $\text{Ces}_p(X)$. We also prove a result on the Opial/uniform Opial property of Cesàro spaces of vector-valued sequences.
The paper studies Banach spaces satisfying the Littlewood-Paley-Rubio de Francia property LPR_p, 2 \leq p < \infty. The paper shows that every Banach lattice whose 2-concavification is a UMD Banach lattice has this property. The paper also shows that every space having LPR_q also has LPR_p with q \leq p < \infty.
Let $\mathcal A$ be a separable Banach algebra, $G$ be a locally compact Hausdorff group and $1< p<\infty$. In this paper, we first provide a necessary and sufficient condition, for which $L^p(G,\mathcal A)$ is a Banach algebra, under convolution product. Then we characterize the character space of $L^p(G,\mathcal A)$, in the case where $\mathcal A$ is commutative and $G$ is abelian. Moreover, we investigate the BSE-property for $L^p(G,\mathcal A)$ and prove that $L^p(G,\mathcal A)$ is a BSE-algebra if and only if $\mathcal A$ is a BSE-algebra and $G$ is finite. Finally, we study the BSE-norm …
On the stability of the Bishop's property($β$) under compact perturbations
本文证明了Banach空间上算子的Bishop性质($\beta$)、分解性质($\delta$)及可分解性在拟幂零、紧和代数可交换扰动下保持稳定，推进了局部谱理论中开放问题的研究。
We study the pointwise multiplier property of the characteristic function of the half-space on weighted mixed-norm anisotropic vector-valued function spaces of Bessel potential and Triebel-Lizorkin type.
We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball $B_E$ of a Banach space $E$ lack weak$^*$-strongly exposed points. Consequently, we obtain that some Banach algebras of holomorphic functions on an arbitrary Banach space have the Daugavet property which extends the observation of P. Wojtaszczyk. Moreover, we present a new denseness result by proving that the set of norm-attaining vector-valued holomorphic functions on the open unit ball of a dual Banach space is dense provided that its predual space has the metric $π$-property. Besid…
We provide a short proof for the twisted multiplicativity property of the operator-valued S-transform. This is my contribution to the topical collection ''Multivariable Operator Theory. The Jörg Eschmeier Memorial'' of Complex Analysis and Operator Theory.
The property of isotonicity of a continuous convex function defined on the entire space or only on the positive cone is characterized via subdifferentials. Numerous examples illustrating the obtained results are included.
Weak stationarity of a matrix valued differential form at superdensity points of its vanishing set
该研究证明了矩阵值微分形式在其零集超密度点处的弱平稳性，并将这一结果应用于Maurer-Cartan方程相关研究，为相关数学分析提供了理论支撑。
For metric spaces $X$ and $Y$, normed spaces $E$ and $F$, and certain subspaces $A(X,E)$ and $A(Y,F)$ of vector-valued continuous functions, we obtain a complete characterization of linear and bijective maps $T:A(X,E)\to A(Y,F)$ preserving common zeros, that is, maps satisfying the property \setcounter{equation}{15} \label{dub} Z(f)\cap Z(g)\neq \emptyset \Longleftrightarrow Z(Tf)\cap Z(Tg)\neq \emptyset for any $f,g\in A(X,E)$, where $Z(f)=\{x\in X:f(x)=0\}$. Moreover, we provide some examples of subspaces for which the automatic continuity of linear bijections having the property (\ref{dub})…
We investigate the problem of existence of a bounded extension to $C(K)$ of a bounded $c_0(I)$-valued operator $T$ defined on the subalgebra of $C(K)$ induced by a continuous increasing surjection $ϕ:K\to L$, where $K$ and $L$ are compact lines. Generalizations of some of the results of [6] about extension of $c_0$-valued operators are obtained. For instance, we prove that when a bounded extension of $T$ exists then an extension can be obtained with norm at most twice the norm of $T$. Moreover, the class of compact lines $L$ for which the $c_0$-extension property is equivalent to the $c_0(I)$-…
On the BSE- property of vector valued Beurling algebra $L^1(G,ω, \mathcal{A})$
研究向量值Beurling代数$L^1(G,ω, \mathcal{A})$的BSE性质，证明其为BSE代数当且仅当$\mathcal{A}$为BSE代数。
We consider a certain type of geometric properties of Banach spaces, which includes for instance octahedrality, almost squareness, lushness and the Daugavet property. For this type of properties, we obtain a general reduction theorem, which, roughly speaking, states the following: if the property in question is stable under certain finite absolute sums (for example finite $\ell^p$-sums), then it is also stable under the formation of corresponding Köthe-Bochner spaces (for example $L^p$-Bochner spaces). From this general theorem, we obtain as corollaries a number of new results as well as some …
It is known that for Banach valued functions there are several approaches to define a Sobolev class. We compare the usual definition via weak derivatives with the Reshetnyak-Sobolev space and with the Newtonian space; in particular, we provide sufficient conditions when all three agree. As well we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it acting as a superposition operator.
Let $X$ be a Banach space $E$ a Köthe function space that does not contain $c_0$. It is shown that the vector valued function space $E(X)$ has the Near Radon Nikodym property if and only if $X$ does.
In this paper we investigate a Gaussian average property of Banach spaces. This property is weaker than the Gordon Lewis property but closely related to this and other unconditional structures. It is also shown that this property implies that certain Hilbert space valued operators defined on subspaces of the given space can be extended.
Interpolation on Convex-set Valued Lebesgue Spaces and its Applications
本文在凸集值Lebesgue空间上证明了两个插值定理，推广了经典插值定理，并据此得到了凸集值分数平均算子与极大算子的有界性及矩阵权重的逆向分解性质。
On the boundedness and Schatten class property of noncommutative martingale paraproducts and operator-valued commutators
本文研究了半交换与纯非交换鞅仿积的 Schatten 类性质，并利用 Hytönen 二进鞅技术与复中位数方法，刻画了涉及一般奇异积分算子的算子值交换子的有界性与 Schatten 类隶属关系，解决了其刻画问题。
Stability results for norm attaining operators with range in vector-valued continuous function spaces
本文研究算子的Bishop-Phelps-Bollobás性质（BPBp），通过刻画相关的广义近似超平面级数性质（广义AHSP），证明了两类关于BPBp和广义AHSP的稳定性结论，丰富了Banach空间上算子逼近性质的研究。
In this work we study the vector-valued Hardy spaces H p (D; F) (1 \leq p \leq \infty) and their relationship with RNP, ARNP and the UMDP properties. By following the approach of Taylor in the scalar-valued case, we prove that, when F and F have the ARNP property, then H p (D; F) and H q (D; F) are canonically topologically isomorphic (for p, q \in (1, \infty) conjugate indices) if and only if F has the UMDP.
Vector-Valued Hardy--Littlewood Maximal Inequalities on Ball Banach Function Spaces
本文通过向量值极大不等式刻画非中心哈代-利特尔伍德极大算子在球巴拿赫函数空间及其Köthe伴随空间上的有界性，所得结果补充了Nieraeth的近期研究成果。
A characterisation of the Daugavet property in spaces of vector-valued Lipschitz functions
本文研究了向量值 Lipschitz 函数空间的结构，证明了长度、非一致离散或无界度量空间下相关空间的 Daugavet 性质与八面体性质，并刻画了使算子空间恒为八面体的 Banach 空间，解决了多个公开问题。
Let $ \mathcal{A} $ be a commutative and semisimple Banach algebra with identity norm one and $ G $ be an abelian locally compact Hausdorff group. In this paper, we study BSE-Property for $L^1(G,\mathcal A)$ and show that $L^1(G,\mathcal A)$ is a BSE algebra if and only if $\mathcal A$ is so.
In this paper, we will obtain the strong type and weak type estimates for vector-valued analogues of intrinsic square functions in the generalized weighted Morrey spaces $M^{Φ,φ}_{w}(\mathbb{R}^n)$. We study the boundedness of intrinsic square functions including the Lusin area integral, Littlewood-Paley $\mathrm{g}$-function and $\mathrm{g}_λ^{*}$ -function and their commutators on vector-valued generalized weighted Morrey spaces $M^{Φ,φ}_{w}(l_2)$. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $φ(x,r)$ without assuming …
In this paper, we are interested in giving two characterizations for the so-called property {\bf L}$_{o,o}$, a local vector valued Bollobás type theorem. We say that $(X, Y)$ has this property whenever given $\eps > 0$ and an operador $T: X \rightarrow Y$, there is $η= η(\eps, T)$ such that if $x$ satisfies $\|T(x)\| > 1 - η$, then there exists $x_0 \in S_X$ such that $x_0 \approx x$ and $T$ itself attains its norm at $x_0$. This can be seen as a strong (although local) Bollobás theorem for operators. We prove that the pair $(X, Y)$ has the {\bf L}$_{o,o}$ for compact operators if and only if …
In this paper we show the weak Banach-Saks property of the Banach vector space $(L_μ^p)^m$ generated by $m$ $L_μ^p$-spaces for $1\leq p<+\infty,$ where $m$ is any given natural number. When $m=1,$ this is the famous Banach-Saks-Szlenk theorem. By use of this property, we also present inequalities for integrals of functions that are the composition of nonnegative continuous convex functions on a convex set of a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of the space $(L_μ^p)^m$ for $1\leq p<+\infty$ and inequalities when these vector-valued functions are in …
We give a new treatment of Quiggin's and McCullough's characterization of complete Nevanlinna-Pick kernels. We show that a kernel has the matrix-valued Nevanlinna-Pick property if and only if it has the vector-valued Nevanlinna-Pick property. We give a representation of all complete Nevanlinna-Pick kernels, and show that they are all restrictions of a universal complete Nevanlinna-Pick kernel.
In this paper, with the aid of the simplicial approximation property, the Hopf's construction and Dugundji's homotopy extension Theorem, we first show that if C is a nonempty compact convex subset of an F-space (E; || ||); then for every positive real number epsilon and every subset D of E containing C and every continuous map f from D to C there exists a continuous map g from D to C which is epsilon-near to f and has only a finite number of fixed points. Secondly, by using this result and the simplicial approximation property, we establish that for any positive real number epsilon and every p…
We give a sufficient condition for a Banach space with which the homogeneous extension of a surjective isometry from the unit sphere of it onto another one is real-linear. The condition is satisfied by a uniform algebra and a certain extremely $C$-regular space of real-valued continuous functions.
We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, `vector-valued' means `taking values in a function lattice with the UMD (unconditional martingale differences) property'. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local…
Suppose $(X,Σ,μ)$ is a finite measure space, $E$ is a Banach lattice, and $B(X,E,μ)$ is the space of all Bochner integrable $E$-valued functions. In this note, we show that $B(X,E,μ)$ is a $KB$-space or has the sequential Fatou property if and only if so is $E$. Among this, some results about Bochner integral convergence in $B(X,E,μ)$, using order structure of $E$, have been proved, as well.
We prove a version of the Lebesgue Differentiation Theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon-Nikodým property.
For a compact Hausdorff space $S$, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on $S$, denoted by $C(S)$, satisfies property-$(P_1)$ (the set-valued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of $C(S)$. Various stability results related to property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps are also established. As a consequence, we derive that if $Y$ is a proximinal finite co-dimensional subspace of $c_0$ then the closed unit ball of $Y$ sat…
In this article, we give new results in the startpoint theory for quasi-pseudometric spaces. The results we present provide us with the existence of startpoint (endpoint, fixed point) for multi-valued maps defined on a bicomplete quasi-pseudometric space. We characterise the existence of startpoint and endpoint by the so-called \textit{mix-point property}. The present results extend known ones in the area.
In this paper we prove a randomized difference norm characterization for Bessel potential spaces with values in UMD Banach spaces. The main ingredients are $\mathcal{R}$-boundedness results for Fourier multiplier operators, which are of independent interest. As an application we characterize the pointwise multiplier property of the indicator function of the half-space on these spaces. All results are proved in the setting of weighted spaces.
Let $X$ be a Banach space and $(Ω,Σ)$ be a measure space. We provide a characterization of sequences in the space of $X$-valued countably additive measures on $Ω,Σ)$ of bounded variation that generate complemented copies of $\ell_1$. As application, we prove that if a dual Banach space $E^*$ has Pełczyński's property (V*) then so does the space of $E^*$-valued countably additive measures with bounded variation. Another application, we show that for a Banach space $X$, the space $\ell_\infty(X)$ contains a complemented copy of $\ell_1$ if and only if $X$ contains all $\ell_1^n$ uniformly comple…
Let $A$ and $B$ be commutative semisimple Banach algebras. In this paper, the $BSE$ and $BED$- property of tensor Banach algebra $A{\otimes}_γ B$ with respect to the Banach algebras $A$ and $B$ are assesed. In particular, $BSE$ and $BED$- structure of vector-valued group algebra $L_{1}(G, A)$ and $C^{*}$- algebra $C_{0}(X, A)$ characterized.
We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset $Ω\subset\mathbb{R}^N$ and a Banach space $V$, we compare the classical Sobolev space $W^{1,p}(Ω, V)$ with the so-called Sobolev-Reshetnyak space $R^{1,p}(Ω, V)$. We see that, in general, $W^{1,p}(Ω, V)$ is a closed subspace of $R^{1,p}(Ω, V)$. As a main result, we obtain that $W^{1,p}(Ω, V)=R^{1,p}(Ω, V)$ if, and only if, the Banach space $V$ has the Radon-Nikodým property
Let $\mathrm{Lip}_0(X)$ be the space of all Lipschitz scalar-valued functions on a pointed metric space $X$. We characterize the approximation property for $\mathrm{Lip}_0(X)$ with the bounded weak* topology using as tools the tensor product, the $ε$-product and the linearization of Lipschitz mappings.
Banach-valued graph limits: Graphon representability and Banach-space structure
该研究关联带巴拿赫装饰的图的图极限问题与巴拿赫空间结构，揭示了不同类型巴拿赫空间满足对应图子表示性质的充要条件。
Affine Approximation in Finite Nagata Dimension and Applications to Lipschitz-free spaces
本文证明若度量空间M的Nagata维数不超过d，则存在以R^d为模型的图册，使得任意Lipschitz映射可被仿射映射一致逼近，并应用于构造ACUG结构及证明Lipschitz自由空间具有Pelczyński性质(V*)。
We investigate the transition semigroup of the solution to a stochastic evolution equation $dX(t) = AX(t)dt +dW_H(t)$, $t\ge 0,$ where $A$ is the generator of a $C_0$-semigroup $S$ on a separable real Banach space $E$ and $W_H$ is cylindrical white noise with values in a real Hilbert space $H$ which is continuously embedded in $E$. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of $S$ in $H$. In particular we investigate the interplay between analyticity of the transition semigr…
The operator space $\text{OUMD}$ property was introduced by Pisier in the context of vector-valued noncommutative $L_p$-spaces. It is an open problem whether the column Hilbert space has this property. Based on some complex interpolation techniques, we are able to relate this problem to the study of the $\text{OUMD}_q$ property for non-commutative $L_p$-spaces.
In this paper, we establish $\mathcal B$-valued variational inequalities for differential operators, ergodic averages and symmetric diffusion semigroups under the condition that Banach space $\mathcal B$ has martingale cotype property. These results generalize, on the one hand Pisier and Xu's result on the variational inequalities for $\mathcal B$-valued martingales, on the other hand many classical variational inequalities in harmonic analysis and ergodic theory. Moreover, we show that Rademacher cotype $q$ is necessary for the $\mathcal B$-valued $q$-variational inequalities. As applications…
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