scholarly journals Strong Convergence of Pramarts in Banach Spaces

1981 ◽  
Vol 33 (2) ◽  
pp. 357-361 ◽  
Author(s):  
Leo Egghe
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Probability Space ◽  
Stopping Times ◽  
Good Property ◽  
Convergence Properties ◽  
Good Convergence ◽  
Image Position ◽  
Sampling Property ◽  
Optional Sampling

Let E be a Banach space and be an adapted sequence on the probability space We denote by T the set of all bounded stopping times with respect to . is called a pramart ifconverges to zero in probability, uniformly in τ ≧ σ. The notion of pramart was introduced in [6]. A good property is the optional sampling property (see Theorem 2.4 in [6]). Furthermore the class of pramarts intersects the class of amarts, and every amart is a pramart if and only if dim E < ∞ ([2], see also [4]). Pramarts behave indeed quite differently than amarts. Although the class of pramarts is large, they have good convergence properties as is seen in the next two results of Millet-Sucheston, [6], [7].THEOREM 1.1. Let be a real-valued pramart of class (d), i.e.,

scholarly journals A strong convergence theorem for contraction semigroups in Banach spaces

2005 ◽  
Vol 72 (3) ◽  
pp. 371-379 ◽  
Author(s):  
Hong-Kun Xu
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Convex Subset ◽  
Convex Banach Space ◽  
Contraction Semigroup ◽  
Strong Convergence Theorem ◽  
Uniformly Convex ◽  
Weakly Continuous ◽  
Duality Map ◽  
Image Position

We establish a Banach space version of a theorem of Suzuki [8]. More precisely we prove that if X is a uniformly convex Banach space with a weakly continuous duality map (for example, lp for 1 < p < ∞), if C is a closed convex subset of X, and if F = {T (t): t ≥ 0} is a contraction semigroup on C such that Fix(F) ≠ ∅, then under certain appropriate assumptions made on the sequences {αn} and {tn} of the parameters, we show that the sequence {xn} implicitly defined byfor all n ≥ 1 converges strongly to a member of Fix(F).

Arret optimal avec contrainte

1983 ◽  
Vol 15 (04) ◽  
pp. 798-812
Author(s):  
Monique Pontier ◽  
Jacques Szpirglas
Keyword(s):  
Probability Space ◽  
Stopping Times ◽  
Average Reward ◽  
Image Position

Given two optional positive bounded processes Y and Y′, defined on a probability space , and a non-negative real a, the problem is to maximize the average reward E(YT ) among all the stopping times T verifying the following constraint: The problem is solved by Lagrangian saddlepoint techniques in the set of randomized stopping times including the set of stopping times.

Arret optimal avec contrainte

1983 ◽  
Vol 15 (4) ◽  
pp. 798-812 ◽  
Author(s):  
Monique Pontier ◽  
Jacques Szpirglas
Keyword(s):  
Probability Space ◽  
Stopping Times ◽  
Average Reward ◽  
Image Position

Given two optional positive bounded processesYandY′, defined on a probability space, and a non-negative reala,the problem is to maximize the average rewardE(YT) among all the stopping timesTverifying the following constraint:The problem is solved by Lagrangian saddlepoint techniques in the set of randomized stopping times including the set of stopping times.

Best L∞-approximation of measurable, vector-valued functions

1984 ◽  
Vol 96 (3) ◽  
pp. 477-481 ◽  
Author(s):  
Abdallah M. Al-Rashed ◽  
Richard B. Darst
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Convex Banach Space ◽  
Equivalence Classes ◽  
Uniformly Convex ◽  
Integrable Functions ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Measurable Vector

Let (Ω, ,μ) be a probability space, and let be a sub-sigma-algebra of . Let X be a uniformly convex Banach space. Let A =L∞(Ω, , μ X) denote the Banach space of (equivalence classes of) essentially bounded μ-Bochner integrable functions g: Ω.→ X, normed by the function ∥.∥∞ defined for g ∈ A by(cf. [6] for a discussion of this space). Let B = L∞(Ω, , μ X), and let f ε A. A sufficient condition for g ε B to be a best L∞-approximation to f by elements of B is established herein.

scholarly journals Some remarks on pramarts and mils

1993 ◽  
Vol 35 (2) ◽  
pp. 239-251 ◽  
Author(s):  
Zhen-Peng Wang ◽  
Xing-Hong Xue
Keyword(s):  
Probability Space ◽  
Stopping Time ◽  
Indicator Function ◽  
Stopping Times ◽  
Norm Topology ◽  
Directed Set ◽  
Stochastic Basis ◽  
Image Position ◽  
The Right ◽  
Directed Sets

Let F be a Banach space, (ω, ℱ, P) a fixed probability space, D a directed set filtering to the right with the order ≤, and (ℱt, D) a stochastic basis of ℱ, i.e. (ℱt, D) is an increasing family of sub-σ-algebras of ℱ:ℱs ⊂ for any s,t ε D and s≤t. Throughout this paper, (Xt) is an F-valued, (ℱt)-adapted sequence, i.e. Xt, is ℱt-measurable, t ε D. We also assume that Xt, ∈ L1, i.e. ∫ ∥Xt∥ <∞. We use I(H) to denote the indicator function of an event H. Let ∞ be a such element: t <∞, t ∈ D, = D ∪ ∞, and ℱ∞ = σ. A stopping time is a map τ:Ω→ such that (τ<t) ∈ ℱt, t ∈ D. A stopping time τ is called simple (countable) if it takes finitely (countably) many values in D(). Let T and Tc be the sets of simple and countable stopping times respectively and Tf = {τ ∈ Tc: τ<∞ a.s.}. Clearly, (T, <) and (Tf, <) are directed sets filtering to the right. For τ ∈ Tc, letand= {(Xt): there is σ∈ Tf such that ∫(ι<∞) ∥Xι∥ < ∞, σ ≤ τ ∈ Tc},= {(Xt):(Xι, ι ∈ T) converges stochastically (i.e. in probability) in the norm topology},ℰ = {(Xt):(Xι, ι ∈ T) converges essentially in the norm topology}.

scholarly journals On ergodic theorem for a Banach valued random sequence

1972 ◽  
Vol 13 (4) ◽  
pp. 501-507 ◽  
Author(s):  
Zoran R. Pop-Stojanovic
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Ergodic Theorem ◽  
Separable Banach Space ◽  
Probability Space ◽  
Random Sequence ◽  
Random Variables ◽  
Strong Dual

In this paper we shall deal with a probability space (S, Σ, P), a separable Banach space X having its strong dual X* and a strictly stationary random sequence defined as in [7], where are X-valued, Gelfand-Pettis (weakly) integrable [6], [9], and strongly measurable random variables. In the case when Yk's are Bochner (strongly) integrable random variables one can find the ergodic theorem for such a sequence and, with respect to strong convergence in X, in the papers [7], [8].

scholarly journals Some results on the span of families of Banach valued independent, random variables

1991 ◽  
Vol 14 (2) ◽  
pp. 381-384
Author(s):  
Rohan Hemasinha
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Linear Span ◽  
Random Variables ◽  
Isomorphic Copy ◽  
Independent Random Variables ◽  
Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

On the centres of hereditary JBW-subalgebras of a JBW-algebra

1979 ◽  
Vol 85 (2) ◽  
pp. 317-324 ◽  
Author(s):  
C. M. Edwards
Keyword(s):  
Banach Space ◽  
General Theory ◽  
Jordan Algebra ◽  
Identity Element ◽  
Image Position

A JB-algebra A is a real Jordan algebra, which is also a Banach space, the norm in which satisfies the conditions thatandfor all elements a and b in A. It follows from (1.1) and (l.2) thatfor all elements a and b in A. When the JB-algebra A possesses an identity element then A is said to be a unital JB-algebra and (1.2) is equivalent to the condition thatfor all elements a and b in A. For the general theory of JB-algebras the reader is referred to (2), (3), (7) and (10).

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