Pointwise Convergence of Alternating Sequences

1988 ◽  
Vol 40 (3) ◽  
pp. 610-632 ◽  
Author(s):  
M. A. Akcoglu ◽  
L. Sucheston
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Pointwise Convergence ◽  
Finite Measure ◽  
Linear Contraction ◽  
Negative Function ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Complex Valued

Let 1 < p < ∞ and let Lp be the usual Banach Space of complex valued functions on a σ-finite measure space. Let (Tn), n ≧ 1, be a sequence of positive linear contractions on Lp. Hence and , where is the part of Lp that consists of non-negative Lp functions. The adjoint of Tn is denoted by which is a positive linear contraction of Lq with q = p/(p — 1).Our purpose in this paper is to show that the alternating sequences associated with (Tn), as introduced in [2], converge almost everywhere. Complete definitions will be given later. When applied to a non negative function, however, this result is reduced to the following theorem.(1.1) THEOREM. If (Tn) is a sequence of positive contractions of Lp then (1.2) exists a.e. for all.

scholarly journals Isometries of measurable functions

1981 ◽  
Vol 24 (1) ◽  
pp. 13-26 ◽  
Author(s):  
Michael Cambern
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Finite Measure ◽  
Measurable Operator ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Linear Isometries

Let (X, Σ, μ) be a σ-finite measure space and denote by L∞(X, K) the Banach space of essentially bounded, measurable functions F defined on X and taking values in a separable Hilbert space K. In this article a characterization is given of the linear isometries of L∞(X, K) onto itself. It is shown that if T is such an isometry then T is of the form (T(F))(x) = U(x)(φ(F))(x), where φ is a set isomorphism of Σ onto itself, and U is a measurable operator-valued function such that U(x) is almost everywhere an isometry of K onto itself. It is a consequence of the proof given here that every isometry of L∞(X, K) is the adjoint of an isometry of L1(x, K).

scholarly journals POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 84 (1) ◽  
pp. 44-48 ◽  
Author(s):  
MICHAEL G. COWLING ◽  
MICHAEL LEINERT
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Semigroup Of Operators ◽  
Almost Everywhere ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

scholarly journals Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

scholarly journals On the almost everywhere convergence of the ergodic averages

1990 ◽  
Vol 10 (1) ◽  
pp. 141-149
Author(s):  
F. J. Martín-Reyes ◽  
A. De La Torre
Keyword(s):  
Measure Space ◽  
Uniform Boundedness ◽  
Finite Measure ◽  
Necessary And Sufficient Condition ◽  
Ergodic Averages ◽  
Measurable Transformation ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Finite Invariant Measure ◽  
Necessary And Sufficient

AbstractLet (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

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.

Akcoglu's Ergodic Theorem for Uniform Sequences

1980 ◽  
Vol 32 (4) ◽  
pp. 880-884
Author(s):  
James H. Olsen
Keyword(s):  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Positive Contraction ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Image Position

Let (X, F,) be a sigma-finite measure space. In what follows we assume p fixed, 1 < p < ∞ . Let T be a contraction of Lp(X, F, μ) (‖T‖,p ≦ 1). If ƒ ≧ 0 implies Tƒ ≧ 0 we will say that T is positive. In this paper we prove that if is a uniform sequence (see Section 2 for definition) and T is a positive contraction of Lp, thenexists and is finite almost everywhere for every ƒ ∊ Lp(X, F, μ).

The Individual Ergodic Theorem for Contractions with Fixed Points

1980 ◽  
Vol 23 (1) ◽  
pp. 115-116 ◽  
Author(s):  
James H. Olsen
Keyword(s):  
Fixed Points ◽  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Individual Ergodic Theorem ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
The Individual ◽  
Image Position

Let (X, I, μ) be a σ-finite measure space and let T take Lp to Lp, p fixed, 1<p<∞,‖t‖p≤1. We shall say that the individual ergodic theorem holds for T if for any uniform sequence K1, k2,… (for the definition, see [2]) and for any f∊LP(X), the limitexists and is finite almost everywhere.

scholarly journals Best N-Simultaneous Approximation in Lp(μ,X)

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
Tijani Pakhrou
Keyword(s):  
Banach Space ◽  
Compact Subset ◽  
Measure Space ◽  
Simultaneous Approximation ◽  
Finite Measure ◽  
Weakly Compact ◽  
Integrable Functions ◽  
Finite Measure Space

Let X be a Banach space. Let 1≤p<∞ and denote by Lp(μ,X) the Banach space of all X-valued Bochner p-integrable functions on a certain positive complete σ-finite measure space (Ω,Σ,μ), endowed with the usual p-norm. In this paper, the theory of lifting is used to prove that, for any weakly compact subset W of X, the set Lp(μ,W) is N-simultaneously proximinal in Lp(μ,X) for any arbitrary monotonous norm N in Rn.

An Extrapolation Theorem for Positive Operators

1978 ◽  
Vol 30 (02) ◽  
pp. 225-230
Author(s):  
H. D. B. Miller
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Positive Operators ◽  
Vector Spaces ◽  
Complex Vector ◽  
Linear Map ◽  
Positive Linear Map ◽  
Finite Measure Space ◽  
Extrapolation Theorem ◽  
Complex Valued

Denote by S and M respectively the complex vector spaces of simple and measurable complex valued functions defined on the finite measure space X. Let T be a positive linear map from S to M such that for each p, 1 &lt; p &lt; ∞, sup {||T f||p: f ∈ S, ||f||P ≦ 1} is finit. finite. T then has an extension to a bounded transformation of every LP(X), 1 &lt; p &lt; ∞ , and these extensions are "consistent". The norm of T as a transformation of Lp is denoted ||T||P. The aim of this note is to prove the following theorem.

scholarly journals Best approximation in Orlicz spaces

1991 ◽  
Vol 14 (2) ◽  
pp. 245-252 ◽  
Author(s):  
H. Al-Minawi ◽  
S. Ayesh
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Best Approximation ◽  
Measure Space ◽  
Real Banach Space ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Finite Measure ◽  
Reflexive Subspace ◽  
Finite Measure Space

LetXbe a real Banach space and(Ω,μ)be a finite measure space andϕbe a strictly icreasing convex continuous function on[0,∞)withϕ(0)=0. The spaceLϕ(μ,X)is the set of all measurable functionsfwith values inXsuch that∫Ωϕ(‖c−1f(t)‖)dμ(t)<∞for somec>0. One of the main results of this paper is: “For a closed subspaceYofX,Lϕ(μ,Y)is proximinal inLϕ(μ,X)if and only ifL1(μ,Y)is proximinal inL1(μ,X)′​′. As a result ifYis reflexive subspace ofX, thenLϕ(ϕ,Y)is proximinal inLϕ(μ,X). Other results on proximinality of subspaces ofLϕ(μ,X)are proved.

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