scholarly journals Invariant and Absolute Invariant Means of Double Sequences

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Abdullah Alotaibi ◽  
M. Mursaleen ◽  
M. A. Alghamdi
Keyword(s):  
Invariant Mean ◽  
Double Sequences ◽  
Sublinear Functionals ◽  
Invariant Means ◽  
The Absolute

We examine some properties of the invariant mean, define the concepts of strongσ-convergence and absoluteσ-convergence for double sequences, and determine the associated sublinear functionals. We also define the absolute invariant mean through which the space of absolutelyσ-convergent double sequences is characterized.

The Extensions of an Invariant Mean and the Set LIM ∽ TLIM

1994 ◽  
Vol 46 (4) ◽  
pp. 808-817
Author(s):  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Discrete Group ◽  
Locally Compact Group ◽  
Invariant Mean ◽  
Locally Compact ◽  
Invariant Means

AbstractLet with . If G is a nondiscrete locally compact group which is amenable as a discrete group and m ∈ LIM(CB(G)), then we can embed into the set of all extensions of m to left invariant means on L∞(G) which are mutually singular to every element of TLIM(L∞(G)), where LIM(S) and TLIM(S) are the sets of left invariant means and topologically left invariant means on S with S = CB(G) or L∞(G). It follows that the cardinalities of LIM(L∞(G)) ̴ TLIM(L∞(G)) and LIM(L∞(G)) are equal. Note that which contains is a very big set. We also embed into the set of all left invariant means on CB(G) which are mutually singular to every element of TLIM(CB(G)) for G = G1 ⨯ G2, where G1 is nondiscrete, non–compact, σ–compact and amenable as a discrete group and G2 is any amenable locally compact group. The extension of any left invariant mean on UCB(G) to CB(G) is discussed. We also provide an answer to a problem raised by Rosenblatt.

Some new $$I$$ I -convergent double sequences space of invariant means

2015 ◽  
Vol 26 (7-8) ◽  
pp. 1697-1708
Author(s):  
Vakeel A. Khan ◽  
Sabiha Tabassum ◽  
Nazneen Khan
Keyword(s):  
Double Sequences ◽  
Invariant Means

scholarly journals INVARIANT MEAN AND SOME CORE THEOREMS FOR DOUBLE SEQUENCES

2010 ◽  
Vol 14 (1) ◽  
pp. 21-33 ◽  
Author(s):  
M. Mursaleen ◽  
S. A. Mohiuddine
Keyword(s):  
Invariant Mean ◽  
Double Sequences

On Invariant Means which are Not Inverse Invariant

1968 ◽  
Vol 20 ◽  
pp. 222-224
Author(s):  
M. Rajagopalan ◽  
K. G. Witz
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Torsion Group ◽  
Invariant Mean ◽  
Invariant Means ◽  
Arbitrary Abelian Group

In (1) R. G. Douglas says: “For a finite abelian group there exists a unique invariant mean which must be inversion invariant. For an infinite torsion abelian group it is not clear what the situation is.” It is not hard to see that if every element of an abelian group G is of order 2, then every invariant mean on G is also inversion invariant (see 1, remark 4). In this note we prove the following theorem (Theorem 1 below): An abelian torsion group G has an invariant mean which is not inverse invariant if, and only if, 2G is infinite. This result, together with the theorems of Douglas, answers completely the question of the existence (on an arbitrary abelian group) of invariant means which are not inverse invariant.

scholarly journals Invariant Means on Banach Spaces

2017 ◽  
Vol 31 (1) ◽  
pp. 127-140
Author(s):  
Radosław Łukasik
Keyword(s):  
Banach Spaces ◽  
Invariant Mean ◽  
Sufficient Condition ◽  
Invariant Means

Abstract In this paper we study some generalization of invariant means on Banach spaces. We give some sufficient condition for the existence of the invariant mean and some examples where we have not it.

scholarly journals On Some Lacunary Almost Convergent Double Sequence Spaces and Banach Limits

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Metin Başarır ◽  
Şükran Konca
Keyword(s):  
Double Sequence ◽  
Sequence Spaces ◽  
Almost Convergence ◽  
Double Sequences ◽  
Double Sequence Spaces ◽  
Inclusion Relations ◽  
Banach Limits ◽  
Sublinear Functionals

The object of this paper is to introduce some new sequence spaces related with the concept of lacunary strong almost convergence for double sequences and also to characterize these spaces through sublinear functionals that both dominate and generate Banach limits and to establish some inclusion relations.

scholarly journals Invariant linear functionals

1972 ◽  
Vol 14 (3) ◽  
pp. 293-303
Author(s):  
W. Randolph Woodward ◽  
R. R. Chivukula
Keyword(s):  
Banach Space ◽  
Supremum Norm ◽  
Invariant Mean ◽  
Semigroup Of Operators ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Multiplicative Structure ◽  
Continuous Linear Functional ◽  
Comprehensive Survey ◽  
Invariant Means

Let B be a Banach space and let ℒ(B) denote the space of all bounded inear operators from B to B, which is a Banach algebra under composition of operators as multiplication. By a semigroup of operators G on B, we mean a norm bounded subser G of ℒ (B) which is a subsemigroup in the multiplicative structure of ℒ(B). The purpose of this paper is to study the existence of nonzero continuous linear functionals on B invariant under G, that is given B and G, does there exist μ∈B*, with μ ≠ 0, such that μ(Sx) = μ(x) for all x∈B and S∈G. This question is an attempt to generalize the familiar concepts of invariant means and amenability of semigroups. If H is any semigroup and m(H) is the Banach space of all bounded real valued functions on H with supremum norm, then a mean is a positive normalized continuous linear functional on m(H). A mean is called (left) [right] invariant if it is invariant under (left) [right] translations and H is called (left)[right] amenable if there exists such a mean; e.g., F is a left invariant mean if F∈m(H)* is such that (i) ‖F‖ = 1, (ii) F(x)≧0 if x(g)≧0 for all g∈H, (iii) F(xg) = F(x) for all x∈m(H) and g∈G where xg(h) = x(gh) for all h∈H. Amenability of semigroups has been studied extensively in recent years, for example see Day [2] or Hewitt and Ross [6] for an introduction, and Day [3] for a comprehensive survey.

scholarly journals Statistical σ-convergence of double sequences with application

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2783-2792
Author(s):  
M. Mursaleen ◽  
Cemal Belen ◽  
Syed Rizvi
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Invariant Mean ◽  
Double Sequences ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Functions Of Two Variables

The concepts of ?-statistical convergence, statistical ?-convergence and strong ?q-convergence of single (ordinary) sequences have been introduced and studied in [M. Mursaleen, O.H.H. Edely, On the invariant mean and statistical convergence, App. Math. Lett. 22, (2011), 1700-1704] which were obtained by unifying the notions of density and invariant mean. In this paper, we extend these ideas from single to double sequences. We also use the concept of statistical ?-convergence of double sequences to prove a Korovkin-type approximation theorem for functions of two variables and give an example to show that our Korovkin-type approximation theorem is stronger than those proved earlier by other authors.

Almost Convergence, Summability And Ergodicity

1974 ◽  
Vol 26 (02) ◽  
pp. 372-387 ◽  
Author(s):  
J. Peter Duran
Keyword(s):  
Summability Method ◽  
Invariant Mean ◽  
Almost Convergence ◽  
Shift Transformation ◽  
Original Definition ◽  
Invariant Means ◽  
Definition Of

The notion of almost convergence introduced by Lorentz [15] has been generalized in several directions (see, for example [1; 8; 11 ; 14; 17]). I t is the purpose of this paper to give a generalization based on the original definition in terms of invariant means. This is effected by replacing the shift transformation by an "ergodic" semigroupof positive regular matrices in the definition of invariant mean. The resulting "- invariant means" give rise to a summability method which we dub-almost convergence.

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