scholarly journals Integration in Orlicz-Bochner Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Marian Nowak
Keyword(s):  
Integral Representation ◽  
Measure Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Continuous Linear ◽  
Topological Properties ◽  
Young Function ◽  
Finite Measure Space ◽  
Bochner Space ◽  
Continuous Linear Operators

Let (Ω,Σ,μ) be a complete σ-finite measure space, φ be a Young function, and X and Y be Banach spaces. Let Lφ(X) denote the Orlicz-Bochner space, and Tφ∧ denote the finest Lebesgue topology on Lφ(X). We study the problem of integral representation of (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y with respect to the representing operator-valued measures. The relationships between (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y and the topological properties of their representing operator measures are established.

scholarly journals Essential supremum norm differentiability

1985 ◽  
Vol 8 (3) ◽  
pp. 433-439
Author(s):  
I. E. Leonard ◽  
K. F. Taylor
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Real Banach Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Supremum Norm ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Fréchet Differentiability ◽  
Finite Measure Space

The points of Gateaux and Fréchet differentiability inL∞(μ,X)are obtained, where(Ω,∑,μ)is a finite measure space andXis a real Banach space. An application of these results is given to the spaceB(L1(μ,ℝ),X)of all bounded linear operators fromL1(μ,ℝ)intoX.

Continuous linear operators on Orlicz-Bochner spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1147-1155 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Finite Measure ◽  
Compact Operators ◽  
Linear Operators ◽  
Continuous Linear ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Completely Continuous Operators ◽  
Weakly Compact Operators ◽  
Continuous Operators ◽  
Continuous Linear Operators

Abstract Let (Ω, Σ, μ) be a complete σ-finite measure space, φ a Young function and X and Y be Banach spaces. Let Lφ(X) denote the corresponding Orlicz-Bochner space and $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$ denote the finest Lebesgue topology on Lφ(X). We examine different classes of ( $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$, ∥ ⋅ ∥Y)-continuous linear operators T : Lφ(X) → Y: weakly compact operators, order-weakly compact operators, weakly completely continuous operators, completely continuous operators and compact operators. The relationships among these classes of operators are established.

scholarly journals Points of narrowness and uniformly narrow operators

2017 ◽  
Vol 9 (1) ◽  
pp. 37-47 ◽  
Author(s):  
A.I. Gumenchuk ◽  
I.V. Krasikova ◽  
M.M. Popov
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Measure Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Standard Tool ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators ◽  
Continuous Linear Operators

It is known that the sum of every two narrow operators on $L_1$ is narrow, however the same is false for $L_p$ with $1 < p < \infty$. The present paper continues numerous investigations of the kind. Firstly, we study narrowness of a linear and orthogonally additive operators on Kothe function spaces and Riesz spaces at a fixed point. Theorem 1 asserts that, for every Kothe Banach space $E$ on a finite atomless measure space there exist continuous linear operators $S,T: E \to E$ which are narrow at some fixed point but the sum $S+T$ is not narrow at the same point. Secondly, we introduce and study uniformly narrow pairs of operators $S,T: E \to X$, that is, for every $e \in E$ and every $\varepsilon > 0$ there exists a decomposition $e = e' + e''$ to disjoint elements such that $\|S(e') - S(e'')\| < \varepsilon$ and $\|T(e') - T(e'')\| < \varepsilon$. The standard tool in the literature to prove the narrowness of the sum of two narrow operators $S+T$ is to show that the pair $S,T$ is uniformly narrow. We study the question of whether every pair of narrow operators with narrow sum is uniformly narrow. Having no counterexample, we prove several theorems showing that the answer is affirmative for some partial cases.

A Multiple Sequence Ergodic Theorem

1983 ◽  
Vol 26 (4) ◽  
pp. 493-497 ◽  
Author(s):  
James H. Olsen
Keyword(s):  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Multiple Sequence ◽  
Finite Measure Space ◽  
Image Position

AbstractLet be a σ-finite measure space, {T1, …, Tk} a set of linear operators of , some p, 1≤p≤∞.Ifexists a.e. for all f ∊ Lp, we say that the multiple sequence ergodic theorem holds for {T1, …, Tk}. If f≥0 implies Tf≥0, we say that T is positive. If there exists an operator S such that |Tf(x)|≥S |f|(x) a.e., we say that T is dominated by S. In this paper we prove that if T1, …, Tk are dominated by positive contractions of , p fixed, 1<p<∞, then the multiple sequence ergodic theorem holds for {T1, …, Tk}.

Symmetric Markovian Semigroups and Dirichlet Forms

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Process ◽  
Dirichlet Form ◽  
Measure Space ◽  
Dirichlet Space ◽  
Finite Measure ◽  
Dirichlet Forms ◽  
Linear Operators ◽  
Dirichlet Spaces ◽  
Finite Measure Space ◽  
Markovian Semigroups

This chapter studies the concepts of Dirichlet form and Dirichlet space by first working with a σ‎-finite measure space (E,B(E),m) without any topological assumption on E and establish the correspondence of the above-mentioned notions to the semigroups of symmetric Markovian linear operators. Later on the chapter assumes that E is a Hausdorff topological space and considers the semigroups and Dirichlet forms generated by symmetric Markovian transition kernels on E. The chapter also considers quasi-regular Dirichlet forms and the quasi-homeomorphism of Dirichlet spaces. From here, the chapter shows that there is a nice Markov process called an m-tight special Borel standard process associated with every quasi-regular Dirichlet form.

Korovkin Theorems for Integral Operators with Kernels of Finite Oscillation

1974 ◽  
Vol 26 (6) ◽  
pp. 1390-1404 ◽  
Author(s):  
M. J. Marsden ◽  
S. D. Riemenschneider
Keyword(s):  
Banach Space ◽  
Banach Lattice ◽  
Measure Space ◽  
Integral Operators ◽  
Finite Measure ◽  
Bounded Sequence ◽  
Positive Operators ◽  
Linear Operators ◽  
Finite Measure Space

There has been considerable interest recently in the investigation of "Korovkin sets". Briefly, for X a Banach space and a family of linear operators on X, a subset K ⊂ X is a Korovkin set relative to if for any bounded sequence {Tn} ⊂ , Tnk → k in X for each k ∊ K implies Tnx → x for each x ∊ X. A large portion of these investigations have been carried out for X being one of the spaces C(S), S compact Hausdorff, the usual Lp spaces of functions on some finite measure space, or some Banach lattice; while is one of the classes +-positive operators, 1-contractions (i.e., ||T|| 1), or + ⋂1

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