Radon-Nikodym Property Research Articles

Noncoercive convex sweeping processes with velocity constraints

In this paper, we investigate noncoercive monotone convex sweeping processes with velocity constraints, a topic not previously investigated. Using some fundamental results on Bochner integration, the Tikhonov regularization method, a solution existence result for coercive convex sweeping processes with velocity constraints and a useful fact on measurable single-valued mappings, we prove the solution existence and properties of the solution set of noncoercive convex sweeping processes with velocity constraints under suitable conditions. These results represent a significant contribution, addressing a specific aspect of an open question posed in our recent work [Adly et al., Convex and nonconvex sweeping processes with velocity constraints: well-posedness and insights. Appl Math Optim. 2023;88(Paper No. 45)]. Additionally, we resolve two other open questions from the same paper concerning the behaviour of the regularized trajectories, relying on the Dominated Convergence Theorem for Bochner integration.

The Adjoint of α-Times-Integrated C-Regularized Semigroups

We consider an operator {S(t)}t≥0 on a Banach space X with generator A, characterized by being an α-times-integrated C-regularized semigroup. The adjoint family S*(t):X*→X* is introduced for analysis. {S*(t)}t≥0 maintains the characteristics of an α-times-integrated C-regularized semigroup, though with strong continuity and Bochner integrals being substituted by weak* continuity and weak* integrals, respectively. Our investigation focuses on the closed subspace X⊙, where {S*(t)}t≥0 exhibits strong continuity. Additionally, a comparison between the adjoint A* of A and the generator of the adjoint family is conducted.

A Radon-Nikodym theorem for monotone measures

A version of Radon-Nikodym theorem for the Choquet integral w.r.t. monotone measures is proved. Without any presumptive condition, we obtain a necessary and sufficient condition for the ordered pair (μ,ν) of finite monotone measures to have the so-called Radon-Nikodym property related to a nonnegative measurable function f. If ν is null-continuous and weakly null-additive, then f is uniquely determined almost everywhere by ν and thus is called the Radon-Nikodym derivative of μ w.r.t. ν. For σ-finite monotone measures, a Radon-Nikodym type theorem is also obtained under the assumption that the monotone measures are lower continuous and null-additive.

Images of Gaussian and other stochastic processes under closed, densely-defined, unbounded linear operators

Gaussian Processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP [Formula: see text] that is the image of another GP [Formula: see text] under a linear transformation [Formula: see text] acting on the sample paths of [Formula: see text] are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when [Formula: see text] is an unbounded operator such as a differential operator, which is common in many modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator [Formula: see text] acting on the sample paths of a square-integrable (not necessarily Gaussian) stochastic process. Our proof technique relies upon Hille’s theorem for the Bochner integral of a Banach-valued random variable.

The dual of a space of compact operators

<abstract><p>Let $ X $ and $ Y $ be Banach spaces. We provide the representation of the dual space of compact operators $ K(X, Y) $ as a subspace of bounded linear operators $ \mathcal{L}(X, Y) $. The main results are: (1) If $ Y $ is separable, then the dual forms of $ K(X, Y) $ can be represented by the integral operator and the elements of $ C[0, 1] $. (2) If $ X^{**} $ has the weak Radon-Nikodym property, then the dual forms of $ K(X, Y) $ can be represented by the trace of some tensor products.</p></abstract>

Hille's Theorem for Bochner Integrals of Functions with Values in Locally Convex Spaces

Hille's Theorem for Bochner Integrals of Functions with Values in Locally Convex Spaces

Geometry of Banach algebra [formula omitted] and the bidual of [formula omitted

This article is intended towards the study of the bidual of generalized group algebra L1(G,A) equipped with two Arens products, where G is any locally compact group and A is a Banach algebra. We show that the left topological center of (L1(G)⊗ˆA)⁎⁎ is a Banach L1(G)-module if G is abelian. Further it also holds permanence property with respect to the unitization of A. We then use this fact to extend the remarkable result of A.M. Lau and V. Losert [7], about the topological center of L1(G)⁎⁎ being just L1(G), to the reflexive Banach algebra valued case using the theory of vector measures. We further explore pseudo-center of L1(G,A)⁎⁎ for non-reflexive Banach algebras A and give a partial characterization for elements of pseudo-center using the Cohen's factorization theorem. In the running we also observe few consequences when A holds the Radon-Nikodym property and weak sequential completeness.

La integral, una visión de su evolución a través del tiempo III

In this opportunity we present some areas of the evolution of the integral, which complement what was deal with in parts I and II. Thus we give a conceptual overview of the McShane integral, of the C-integral, of the Bochner integral, of the Lm HK integral , of the integral on manifolds and on a modern point of view of the Riemann integral.

Statistical Convergence of Spliced Sequences in Terms of Power Series on Topological Spaces

In the present paper, $P-$distributional convergence which is defined by power series method has been introduced. We give equivalent expressions for $P-$distributional convergence of spliced sequences. Moreover, convergence of a bounded $\infty$-spliced sequence via power series method is represented in terms of Bochner integral in Banach spaces.

Generalized Orlicz spaces of Banach-valued functions: Basic theory and duality

For a measure space Ω, we extend the theory of Orlicz spaces generated by an even convex integrand φ:Ω×X→[0,∞] to the case when the range Banach space X is arbitrary. We settle fundamental structural properties such as completeness, characterize separability, reflexivity and represent the dual space. This representation includes the case when X′ has no Radon-Nikodym property or φ is unbounded. We apply our theory to represent convex conjugates and Fenchel-Moreau subdifferentials of integral functionals, leading to the first general such result on function spaces with non-separable range space. For this, we prove a new interchange criterion between infimum and integral for non-separable range spaces, which we consider to be of independent interest.

Statistical Bochner Integral on Frechet Space

Probability theory including the Bochner integral is a very important part of modern mathematical concepts including the modern theory of probabilities, especially in the concept of mathematical expectation and dispersion. In this, study a statistical approach form of Bochner’s theory is given and extended, but some fundamental properties of statistical integral were previously studied in the Banach case. Our approach formulates an extended integration concept of Bochner. By using the statistical convergence on general locally convex space it is possible to obtain very similar results referring to the Frechet space type. From our results, some interesting comparable outputs to Banach space are carried out. At the end of our research, it is conducted that if a function “f” is Bochner integrable in the classic report then it is statistically Bochner integrable, but conversely, this is not true. Hence, the value of the extension of Bochner integration is a need and is the focus of our work. This extension is given by modifying the model published by Schvabik and Guoju. Mathematically it is substantiated that on the space of Frechet types the space of functions of statistical Bochner integrable is a Frechet space.

Absolutely continuous mappings on doubling metric measure spaces

We consider Q-absolutely continuous mappings \(f:X\rightarrow V\) between a doubling metric measure space X and a Banach space V. The relation between these mappings and Sobolev mappings \(f\in N^{1,p}(X;V)\) for \(p\ge Q\ge 1\) is investigated. In particular, a locally Q-absolutely continuous mapping on an Ahlfors Q-regular space is a continuous mapping in \(N^{1,Q}_\textrm{loc}\,(X;V)\), as well as differentiable almost everywhere in terms of Cheeger derivatives provided V satisfies the Radon-Nikodym property. Conversely, though a continuous Sobolev mapping \(f\in N^{1,Q}_\textrm{loc}\,(X;V)\) is generally not locally Q-absolutely continuous, this implication holds if f is further assumed to be pseudomonotone. It follows that pseudomonotone mappings satisfying a relaxed quasiconformality condition are also Q-absolutely continuous.

Fourier transform method for partial differential equations. Part 2. Existence and uniqueness of solutions to the Cauchy problem for linear equations

The article proposes a method for analyzing the Cauchy problem for a wide class of evolutionary linear partial differential equations with variable coefficients. By applying the (inverse) Fourier transform, the original equation is reduced to an integro-differential equation, which can be considered as an ordinary differential equation in the corresponding Banach space. The selection of this space is carried out in such a way that the principle of contraction mappings can be used. To carry out the corresponding estimates for the operators generated by the transformed equation, we impose the conditions of finiteness in the space variable for the inverse Fourier transform of the coefficients, and the spaces of the coefficients of the original equation are determined from the Paley-Wiener Fourier transform theorems. In this case, the apparatus of the theory of the Bochner integral in pseudo-normed spaces, countably-normed spaces and Sobolev spaces is used. Classes of functions are distinguished in which the existence and uniqueness of solutions are proved. For equations with coefficients with separated variables, exact solutions are obtained in the form of a Fourier transform of finite sums for operator exponentials.

Quaternionic exponentially dichotomous operators through S-spectral splitting and applications to Cauchy problem

In this paper, based on the slice regular quaternionic functions and spectral theory on the S-spectrum for quaternionic operators, firstly, the exponential growth bound of quaternionic operator semigroups is studied and the generalized Hille Lemma is obtained through introducing Bochner and Pettis integrals in quaternionic Banach space. Secondly, we study the quaternionic bisemigroups and the direct sum decomposition of quaternionic Banach space, then we introduce the notion of the quaternionic exponentially dichotomous operator and obtain its integral representation formula. It is crucial to note that the quaternionic operators we consider do not necessarily commute. Meanwhile, through establishing the S-spectral splitting theorem, the characterizations of quaternionic exponentially dichotomous operators are demonstrated. Moreover, the slice regular quaternionic bisemigroups generated by the quaternionic bisectorial operator are studied. Thirdly, by introducing the slice quaternionic Banach algebra and multiplicative quaternionic linear functionals, the perturbation invariance of exponential dichotomy for the quaternionic operators is explored. Under the noncommutative setting, our obtained results are essentially different from their analogues in the complex setting. As applications, we consider the Cauchy problems of quaternionic non-homogeneous evolution equations involving the quaternionic semigroup generators and exponentially dichotomous quaternionic operators. Additionally, the slice exponential dichotomy of quaternionic operators is characterized via the corresponding Cauchy problem.

On Complexification of Real Spaces and its Manifestations in the Theory of Bochner and Pettis Integrals

This work is a continuation of our work [14] where we considered linear spaces in the following two situations: a real space admits a multiplication by complex scalars without changing the set itself; a real space is embedded into a wider set with a multiplication by complex scalars. We studied there also how they manifest themselves when the initial space possesses additional structures: topology, norm, inner product, as well as what happens with linear operators acting between such spaces. Changing the linearities of the linear spaces unmasks some very subtle properties which are not thus obvious when the set of scalars is not changed. In the present work we follow the same idea considering now Bochner and Pettis integrals for functions ranged in real and complex Banach and Hilbert spaces. Finally, this leads to the study of strong and weak random elements with values in real and complex Banach and Hilbert spaces, in particular, some properties of their expectations.

Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions

We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. We subsequently use this geometric characterization to answer several questions in Lipschitz analysis. Notably, it follows that the Lipschitz-free spaceF(M)\mathcal {F}(M)over a compact metric spaceMMis a dual space if and only ifMMis purely 1-unrectifiable. Furthermore, we establish a compact determinacy principle for the Radon-Nikodým property (RNP) and deduce that, for any complete metric spaceMM, pure 1-unrectifiability is actually equivalent to some well-known Banach space properties ofF(M)\mathcal {F}(M)such as the RNP and the Schur property. A direct consequence is that any complete, purely 1-unrectifiable metric space isometrically embeds into a Banach space with the RNP. Finally, we provide a possible solution to a problem of Whitney by finding a rectifiability-based description of 1-critical compact metric spaces, and we use this description to prove the following: a bounded turning tree fails to be 1-critical if and only if each of its subarcs hasσ\sigma-finite Hausdorff 1-measure.

Vector Valued Multipliers of McShane Integrable Functions

We study the algebra of vector valued multipliers of Banach algebra valued McShane integrable functions. We prove that if X is a commutative Banach algebra, with identity e of norm one, then functions associated with measures of strong bounded variation and the set {L?([a, b],?) e} are vector valued multipliers of McShane integrable functions. We find some necessary and another set of sufficient conditions for a functiong to define a multiplier. In case X satisfies Radon Nikodym property (weak Radon Nikodym property), we study multiplier operators.

Thick families of geodesics and differentiation

The differentiation theory of Lipschitz functions on metric spaces taking values in a Banach space with the Radon-Nikodým property (RNP), originally developed by Cheeger—Kleiner, has proven to be a powerful tool to prove non-bi-Lipschitz embeddability of metric spaces into these Banach spaces. Important examples of metric spaces to which this theory applies include nonabelian Carnot groups and Laakso spaces. In search of a metric characterization of the RNP, Ostrovskii found another class of spaces that do not bi-Lipschitz embed into RNP spaces, namely spaces containing thick families of geodesics. Our first result is that any metric space containing a thick family of geodesics also contains a subset and a probability measure on that subset which satisfies a weakened form of RNP Lipschitz differentiability. A corollary is a new nonembeddability result. Our second main result is that, if the metric space is a non-RNP Banach space, a subset consisting of a thick family of geodesics can be constructed to satisfy true RNP differentiability. An intriguing question is whether this differentiation criterion, or some weakened form of it such as the one we prove in the first result, actually characterizes general metric spaces non-bi-Lipschitz embeddable into RNP Banach spaces.

Small combination of slices, dentability and stability results of small diameter properties in Banach spaces

In this work we study three different versions of small diameter properties of the unit ball in a Banach space and its dual. The related concepts for all closed bounded convex sets of a Banach space was initiated and developed in [9], [4], [12], [13] was extensively studied in the context of dentability, huskability, Radon Nikodym Property and Krein Milman Property in [14]. We introduce the Ball Huskable Property (BHP), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related properties, BSCSP namely, the unit ball has convex combination of slices of arbitrarily small diameter and BDP namely, the closed unit ball has slices of arbitrarily small diameter. We show BDP implies BHP which in turn implies BSCSP and none of the implications can be reversed. We prove similar results for the w⁎-versions. We prove that all these properties are stable under lp sum for 1≤p≤∞,c0 sum and Lebesgue Bochner spaces. Finally, we explore the stability of these with properties in the light of three space property. We show that BHP is a three space property provided X/Y is finite dimensional and same is true for BSCSP when X has BSCSP and X/Y is strongly regular ([14]).

On the Lattice Structure of the Space of all Bochner Integrable Banach Lattice-Valued Functions

Suppose $(X,\Sigma,\mu)$ is a finite measure space, $E$ is a Banach lattice, and $B(X,E,\mu)$ is the space of all Bochner integrable $E$-valued functions. In this note, we show that $B(X,E,\mu)$ 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 $B(X,E,\mu)$, using order structure of $E$, have been proved, as well.