Radon-Nikodym Property Research Articles

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax–Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.

On parabolic final value problems and well-posedness

We prove that a large class of parabolic final value problems is well posed. This results via explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability of solutions. This data space is the graph normed domain of an unbounded operator, which represents a new compatibility condition pertinent for final value problems. The framework is that of evolution equations for Lax–Milgram operators in vector distribution spaces. The final value heat equation on a smooth open set is also covered, and for non-zero Dirichlet data, a non-trivial extension of the compatibility condition is obtained by addition of an improper Bochner integral.

Mapping theorems for Sobolev spaces of vector-valued functions

We consider Sobolev spaces with values in Banach spaces as they are frequently useful in applied problems. Given two Banach spaces $X\neq\{0\}$ and $Y$, each Lipschitz continuous mapping $F:X\rightarrow Y$ gives rise to a mapping $u\mapsto F\circ u$ from $W^{1,p}(\Omega,X)$ to $W^{1,p}(\Omega,Y)$ if and only if $Y$ has the Radon-Nikodym Property. But if $F$ is one-sided Gateaux differentiable no condition on the space is needed. We also study when weak properties in the sense of duality imply strong properties. Our results are applied to prove embedding theorems, a multi-dimensional version of the Aubin-Lions Lemma and characterizations of the space $W^{1,p}_0(\Omega,X)$.

The dual Radon–Nikodym property for finitely generated Banach C(K)-modules

We extend the well-known criterion of Lotz for the dual Radon–Nikodym property (RNP) of Banach lattices to finitely generated Banach C(K)-modules and Banach C(K)-modules of finite multiplicity. Namely, we prove that if X is a Banach space from one of these classes then its Banach dual $$X^\star $$ has the RNP iff X does not contain a closed subspace isomorphic to $$\ell ^1$$ .

Compactly uniform Bochner integrability of random elements

As the concept of uniform integrability has an essential role in the convergence of moments and martingales in the probability theory it is important to study this concept. In the present paper using Bochner integral we define a new type of uniform integrability for sequences of random elements. We give some independent necessary and sufficient conditions for this concept. We also generalize the concepts of strong and statistical convergences to sequences of random elements and we obtain the relationship between these two important concepts of summability theory via this new type of uniform integrability.

Interpolation des opérateurs de Radon–Nikodym et des espaces L Λ p , h Λ p

Let A 0 , A 1 be two Banach spaces, i:A 0 →A 1 a continuous injection with dense range and 0<α<β<1. In the first part of this work, we show that if i:A α,p →A β,p is a Radon–Nikodym operator, then A α,p has the Radon–Nikodym property. We show that this result is false for complex interpolation (Theorem 2.2). We also show that there are two Banach spaces B 0 ,B 1 and i:B 0 →B 1 a Radon–Nikodym injection such that for 0<α<β<1, i:A α →A β is not a Radon–Nikodym operator. We introduce the interpolation spaces A θ + , θ∈]0,1[, and show that A ¯ θ is an isometric subspace of A θ + .

Bochner integrals in ordered vector spaces

We present a natural way to cover an Archimedean directed ordered vector space E by Banach spaces and extend the notion of Bochner integrability to functions with values in E. The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map.

Lipschitz slices versus linear slices in Banach spaces

The aim of this note is to study the topology generated by Lipschitz slices in the unit sphere of a Banach space. We prove that the above topology agrees with the weak topology in the unit sphere and, as a consequence, we obtain easy Lipschitz characterizations of classical linear topics in Banach spaces as the Daugavet property, Radon-Nikodym property, convex point of continuity property and strong regularity, which shows that the above classical linear properties depend only on the natural uniformity in the Banach space given by the metric and the linear structure.

A new wavelet-like transform associated with the Riesz–Bochner integral and relevant reproducing formula

We introduce a new family of wavelet-like transforms associated to the Riesz–Bochner integral of Lp functions and prove the corresponding reproducing formula of Calderon’s type. Unlike the classical wavelet transforms, this is a class of wavelet-like transforms generated by three components, namely, a kernel function (the Riesz–Bochner integral), a weight function and a wavelet function. The first component depends on many variables, but the second and third ones, which are in our disposal, depend only on one variable and therefore give extra simplicity in applications.

Boundary values of vector-valued Hardy spaces on nonsmooth domains and the Radon–Nikodym property

We define Hardy spaces of functions taking values on a Banach space X over nonsmooth domains. The types of functions we consider are harmonic functions on a starlike Lipschitz domain and solutions to the heat equation on a time-varying domain. Our purpose is twofold: (a) to characterize the Radon–Nikodym property of the Banach space X in terms of the existence of nontangential limits of X-valued functions u in the corresponding Hardy space with index p≥1, (b) to identify the function of the boundary values of u in the Hardy space with index p>1 with an element in the space VXp of measures of p-bounded variation in the absence of the Radon–Nikodym property of X. This extends similar results already known on the unit disk of C and the semispace Rn×(0,∞).

English

We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum \nolimits _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series $\sum \nolimits _{n=1}^{\infty }x_n|E_n|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$. In this paper we give some conditions for variational Henstock integrability of a certain class of such functions.

CONVERGENCE AND ALMOST SURE $T$-STABILITY FOR RANDOM NOOR-TYPE ITERATIVE SCHEME

The purpose of this study is to introduce a Noor-type random iterative scheme and prove the convergence of this kind of random iterative scheme for certain �-weakly con- tractive type random operators. The Bochner integrability of random fixed points for this kind of random operators and the almost sure T -stability and convergence for Noor-type random iterative scheme are established. Our results are stochastic generalizations of the deterministic fixed point theorems of Berinde [7, 8] and Rhoades [29]-[32].

On Spectral Properties in Vector-Valued Beurling Algebra

Let $G$ be an LCA group, $\om$ be a weight function on $G$, and $\A$ be a semisimple, commutative Banach algebra. We characterize some Banach algebra properties of vector-valued Beurling algebra $\Lgwa$ in terms of $\Lgw$ and $\A$ using the Bochner integration theory. These properties are unique uniform norm property (UUNP), unique $C^{\ast}$-norm property (UC$^{\ast}$NP), quasi divisor of zero property (QDZP), weak regularity (WR), regularity, and complete regularity (CR).

The Radon-Nikodym Property and the Limit Average Range

Abstract In this paper we present new characterizations of Banach spaces possessing the Radon-Nikodym property in terms of the limit average range of additive interval functions.

Higher moments of Banach space valued random variables

We define the k:th moment of a Banach space valued random variable as the expectation of its k: th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general. Several chapters are devoted to results in special Banach spaces, including Hilbert spaces, C(K) and D[0,1]. The latter space is non-separable, which complicates the arguments, and we prove various preliminary results on e.g. measurability in D[0,1] that we need. One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix.

Impact of the Radon–Nikodým property on convex approximation properties

A classical result states that the approximation property (AP) of a dual space X* is metric whenever X* or X** has the Radon–Nikodým property (RNP). A version of the proof by Oja deals with the weak metric AP of the original space X and uses lifting techniques to return to the dual space. We streamline this proof, eliminating the need to enter the original space and show that the only essential component of the proof is Oja’s “RNP impact lemma”. We express the weak metric AP and notions of a similar kind in a concise language of systems of seminorms on operator spaces and provide results in the general setting of convex APs.

A NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF B-VALUED RANDOM VARIABLES

Abstract. Inthisarticle,wediscussthecompletemomentconvergencefor arrays of B-valued random variables. We obtain some new resultswhichimprovethecorrespondingonesofSungandVolodin[17]. 1. IntroductionLet {Ω,F,P} be a probability space, and let B be a separable real Banachspace with norm ||·||. A random element is defined to be an F-measurablemapping of Ω into B equipped with the Borel σ-algebra (that is, the σ-algebragenerated by the open sets determined by ||·||). The expected value of a B-valued random element X is defined to be the Bochner integral and denotedby EX.Let {X nk ,k ≥ 1,n ≥ 1} be an array of random elements in a real Banachspace. An array of rowwise random elements {X nk ,k ≥ 1,n ≥ 1} is said to bestochastically dominated by a random variable X (write {X nk } ≺ X) if thereexists a constant C > 0 such thatsup n≥1,k≥1 P(||X nk || > x) ≤ CP(|X| > x), ∀x > 0.Now we recall the following concepts of convergence which were introducedby Hsu and Robbins [6] and Chow [4], respectively.Definition 1.1. A sequence of random variables {X

Numerical approximation of fractional powers of elliptic operators

We present and study a novel numerical algorithm to approximate the action of $T^\beta:=L^{-\beta}$ where $L$ is a symmetric and positive definite unbounded operator on a Hilbert space $H_0$. The numerical method is based on a representation formula for $T^{-\beta}$ in terms of Bochner integrals involving $(I+t^2L)^{-1}$ for $t\in(0,\infty)$. To develop an approximation to $T^\beta$, we introduce a finite element approximation $L_h$ to $L$ and base our approximation to $T^\beta$ on $T_h^\beta:= L_h^{-\beta}$. The direct evaluation of $T_h^{\beta}$ is extremely expensive as it involves expansion in the basis of eigenfunctions for $L_h$. The above mentioned representation formula holds for $T_h^{-\beta}$ and we propose three quadrature approximations denoted generically by $Q_h^\beta$. The two results of this paper bound the errors in the $H_0$ inner product of $T^\beta-T_h^\beta\pi_h$ and $T_h^\beta-Q_h^\beta$ where $\pi_h$ is the $H_0$ orthogonal projection into the finite element space. We note that the evaluation of $Q_h^\beta$ involves application of $(I+(t_i)^2L_h)^{-1}$ with $t_i$ being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. Numerical experiments illustrating the theoretical estimates are provided for both the quadrature error $T_h^\beta-Q_h^\beta$ and the finite element error $T^\beta-T_h^\beta\pi_h$.

Connections Between Metric Characterizations of Superreflexivity and the Radon–Nikodým Property for Dual Banach Spaces

AbstractJohnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon–Nikodým property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any setMwhose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold and thatM=l2is a counterexample.

Inverse Limit Spaces Satisfying a Poincaré Inequality

Abstract We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces, and a large class of new examples. As follows easily from [4], generically our examples have the property that they do not bilipschitz embed in any Banach space with Radon-Nikodym property. For Laakso spaces, thiswas noted in [4]. However according to [7] these spaces admit a bilipschitz embedding in L1. For Laakso spaces, this was announced in [5].