scholarly journals Convolution structures for an Orlicz space with respect to vector measures on a compact group

2021 ◽  
Vol 64 (1) ◽  
pp. 87-98
Author(s):  
Manoj Kumar ◽  
N. Shravan Kumar
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Orlicz Space ◽  
Vector Measure ◽  
Young Function ◽  
Natural Conditions ◽  
Vector Measures ◽  
Convolution Structure

The aim of this paper is to present some results about the space $L^{\varPhi }(\nu ),$ where $\nu$ is a vector measure on a compact (not necessarily abelian) group and $\varPhi$ is a Young function. We show that under natural conditions, the space $L^{\varPhi }(\nu )$ becomes an $L^{1}(G)$-module with respect to the usual convolution of functions. We also define one more convolution structure on $L^{\varPhi }(\nu ).$

scholarly journals On the Radon-Nikodym Property for Vector Measures and Extensions of Transfunctions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Piotr Mikusiński ◽  
John Paul Ward
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Vector Measure ◽  
Measurable Space ◽  
Unique Extension ◽  
Natural Conditions ◽  
Vector Measures ◽  
Nikodym Property

AbstractIf \left( {{\mu _n}} \right)_{n = 1}^\infty are positive measures on a measurable space (X, Σ) and \left( {{v_n}} \right)_{n = 1}^\infty are elements of a Banach space 𝔼 such that \sum\nolimits_{n = 1}^\infty {\left\| {{v_n}} \right\|{\mu _n}\left( X \right)} < \infty, then \omega \left( S \right) = \sum\nolimits_{n = 1}^\infty {{v_n}{\mu _n}\left( S \right)} defines a vector measure of bounded variation on (X, Σ). We show 𝔼 has the Radon-Nikodym property if and only if every 𝔼-valued measure of bounded variation on (X, Σ) is of this form. This characterization of the Radon-Nikodym property leads to a new proof of the Lewis-Stegall theorem.We also use this result to show that under natural conditions an operator defined on positive measures has a unique extension to an operator defined on 𝔼-valued measures for any Banach space 𝔼 that has the Radon-Nikodym property.

A CONVOLUTION-INDUCED TOPOLOGY ON THE ORLICZ SPACE OF A LOCALLY COMPACT GROUP

2015 ◽  
Vol 99 (1) ◽  
pp. 1-11
Author(s):  
IBRAHIM AKBARBAGLU ◽  
SAEID MAGHSOUDI
Keyword(s):  
Compact Group ◽  
Orlicz Space ◽  
Haar Measure ◽  
Locally Compact Group ◽  
Young Function ◽  
Strict Topology ◽  
Locally Compact ◽  
Induced Topology ◽  
Locally Convex Topology ◽  
Locally Convex

Let $G$ be a locally compact group with a fixed left Haar measure. In this paper, given a strictly positive Young function ${\rm\Phi}$, we consider $L^{{\rm\Phi}}(G)$ as a Banach left $L^{1}(G)$-module. Then we equip $L^{{\rm\Phi}}(G)$ with the strict topology induced by $L^{1}(G)$ in the sense of Sentilles and Taylor. Some properties of this locally convex topology and a comparison with weak$^{\ast }$, bounded weak$^{\ast }$ and norm topologies are presented.

scholarly journals Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 326-339
Author(s):  
H.H. Bang ◽  
V.N. Huy
Keyword(s):  
Fourier Transform ◽  
Orlicz Space ◽  
Differential Operators ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Young Function ◽  
The Fourier Transform

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

scholarly journals A note on the duality of locally compact groups

1968 ◽  
Vol 9 (2) ◽  
pp. 87-91 ◽  
Author(s):  
J. W. Baker
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Group Topology ◽  
Locally Compact ◽  
Natural Way

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

Completeness of the L1-space of closed vector measures

1990 ◽  
Vol 33 (1) ◽  
pp. 71-78 ◽  
Author(s):  
Werner J. Ricker
Keyword(s):  
Uniform Convergence ◽  
Convex Space ◽  
Vector Measure ◽  
Sufficient Conditions ◽  
Locally Convex Space ◽  
Vector Measures ◽  
Locally Convex

The notion of a closed vector measure m, due to I. Kluv´;nek, is by now well established. Its importance stems from the fact that if the locally convex space X in which m assumes its values is sequentially complete, then m is closed if and only if its L1-space is complete for the topology of uniform convergence of indefinite integrals. However, there are important examples of X-valued measures where X is not sequentially complete. Sufficient conditions guaranteeing the completeness of L1(m) for closed X-valued measures m are presented without the requirement that X be sequentially complete.

scholarly journals The bounded vector measure associated to a conical measure and pettis differentiability

2001 ◽  
Vol 70 (1) ◽  
pp. 10-36
Author(s):  
L. Rodriguez-Piazza ◽  
M. C. Romero-Moreno
Keyword(s):  
Vector Measure ◽  
Locally Convex Space ◽  
Finite Intersection ◽  
Vector Measures ◽  
Weak Space ◽  
Countably Additive Vector Measure ◽  
Differentiable Vector ◽  
Locally Convex ◽  
Additive Vector ◽  
Additive Vector Measure

AbstractLet X be a locally convex space. Kluvánek associated to each X-valued countably additive vector measure a conical measure on X; this can also be done for finitely additive bounded vector measures. We prove that every conical measure u on X, whose associated zonoform Ku is contained in X, is associated to a bounded additive vector measure σ(u) defined on X, and satisfying σ(u)(H) ∈ H, for every finite intersection H of closed half-spaces. When X is a complete weak space, we prove that σ(u) is countably additive. This allows us to recover two results of Kluvánek: for any X, every conical measure u on it with Ku ⊆ X is associated to a countably additive X-valued vector measure; and every conical measure on a complete weak space is localizable. When X is a Banach space, we prove that σ(u) is countably additive if and only if u is the conical measure associated to a Pettis differentiable vector measure.

scholarly journals How to Obtain Maximal and Minimal Subranges of Two-Dimensional Vector Measures

2019 ◽  
Vol 74 (1) ◽  
pp. 85-90
Author(s):  
Jerzy Legut ◽  
Maciej Wilczyński
Keyword(s):  
Vector Measure ◽  
Measurable Space ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Vector Measures ◽  
Maximal Subset ◽  
Minimal Subset

Abstract Let (X, ℱ) be a measurable space with a nonatomic vector measure µ =(µ1, µ2). Denote by R(Y) the subrange R(Y)= {µ(Z): Z ∈ ℱ, Z ⊆ Y }. For a given p ∈ µ(ℱ) consider a family of measurable subsets ℱp = {Z ∈ ℱ : µ(Z)= p}. Dai and Feinberg proved the existence of a maximal subset Z* ∈ Fp having the maximal subrange R(Z*) and also a minimal subset M* ∈ ℱp with the minimal subrange R(M*). We present a method of obtaining the maximal and the minimal subsets. Hence, we get simple proofs of the results of Dai and Feinberg.

scholarly journals On the stable rank and real rank of group $C^*$-algebras of nilpotent locally compact groups

2005 ◽  
Vol 97 (1) ◽  
pp. 89
Author(s):  
Robert J. Archbold ◽  
Eberhard Kaniuth
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Stable Rank ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Rank ◽  
Locally Compact ◽  
C Algebras ◽  
Necessary And Sufficient

It is shown that if $G$ is an almost connected nilpotent group then the stable rank of $C^*(G)$ is equal to the rank of the abelian group $G/[G,G]$. For a general nilpotent locally compact group $G$, it is shown that finiteness of the rank of $G/[G,G]$ is necessary and sufficient for the finiteness of the stable rank of $C^*(G)$ and also for the finiteness of the real rank of $C^*(G)$.

scholarly journals On copies of the null sequence Banach space in some vector measure spaces

1999 ◽  
Vol 59 (3) ◽  
pp. 443-447
Author(s):  
J.C. Ferrando ◽  
J.M. Amigó
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Vector Measure ◽  
Null Sequence ◽  
Vector Measures ◽  
Measure Spaces ◽  
Additive Vector

In this note we extend a result of Drewnowski concerning copies of C0 in the Banach space of all countably additive vector measures and study some properties of complemented copies of C0 in several Banach spaces of vector measures.

Large deviations for random evolution equation in Besov-Orlicz space

2020 ◽  
pp. 357-387
Author(s):  
Dina Miora Rakotonirina ◽  
Jocelyn Hajaniaina Andriatahina ◽  
Rado Abraham Randrianomenjanahary ◽  
Toussaint Joseph Rabeherimanana
Keyword(s):  
Evolution Equation ◽  
Large Deviations ◽  
Orlicz Space ◽  
Evolution Equations ◽  
Random Evolution ◽  
Young Function ◽  
Large Deviations Principle

In this paper, we develop a large deviations principle for random evolution equations to the Besov-Orlicz space $\mathcal{B}_{M_2, w}^{v, 0}$ corresponding to the Young function $M_2(x)=\exp(x^2)-1$.

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