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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 465364, 9 pages
http://dx.doi.org/10.1155/2012/465364
Research Article

Invariant and Absolute Invariant Means of Double Sequences

1Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 20 February 2012; Revised 17 April 2012; Accepted 28 April 2012

Academic Editor: Sivaram K.Β Narayan

Copyright Β© 2012 Abdullah Alotaibi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

1. Introduction and Preliminaries

For the following notions, we refer to [1, 2].

A double sequence π‘₯=(π‘₯π‘—π‘˜) of real or complex numbers is said to be bounded ifβ€–π‘₯β€–βˆž=sup𝑗,π‘˜||π‘₯π‘—π‘˜||<∞.(1.1) The space of all bounded double sequences is denoted by ℳ𝑒.

A double sequence π‘₯=(π‘₯π‘—π‘˜) is said to converge to the limit L in Pringsheim’s sense (shortly, p-convergent to L) if for every πœ€>0 there exists an integer 𝑁 such that |π‘₯π‘—π‘˜βˆ’πΏ|<πœ€ whenever 𝑗,π‘˜>𝑁. In this case 𝐿 is called the 𝑝-limit of π‘₯. If in addition π‘₯βˆˆβ„³π‘’, then π‘₯ is said to be boundedly convergent to L in Pringsheim’s sense (shortly, bp-convergent to L).

A double sequence π‘₯=(π‘₯π‘—π‘˜) is said to converge regularly to L (shortly, r-convergent to L) if π‘₯ is 𝑝-convergent and the limits π‘₯π‘—βˆΆ=limπ‘˜π‘₯π‘—π‘˜(π‘—βˆˆβ„•) and π‘₯π‘˜βˆΆ=lim𝑗π‘₯π‘—π‘˜(π‘˜βˆˆβ„•) exist. Note that in this case the limits lim𝑗limπ‘˜π‘₯π‘—π‘˜ and limπ‘˜lim𝑗π‘₯π‘—π‘˜ exist and are equal to the 𝑝-limit of π‘₯.

In general, for any notion of convergence 𝜈, the space of all 𝜈-convergent double sequences will be denoted by π’žπœˆ and the limit of a 𝜈-convergent double sequence π‘₯ by 𝜈-lim𝑗,π‘˜π‘₯π‘—π‘˜, where 𝜈∈{𝑝,𝑏𝑝,π‘Ÿ}.

Let Ξ© denote the vector space of all double sequences with the vector space operations defined coordinatewise. Vector subspaces of Ξ© are called double sequence spaces.

All considered double sequence spaces are supposed to containξ€½πžspan𝐣𝐀,βˆ£π‘—,π‘˜βˆˆβ„•(1.2) where𝐞𝐣𝐀𝐒π₯=ξ‚»1,if(𝑗,π‘˜)=(𝑖,β„“),0,otherwise.(1.3)

We denote the pointwise sums βˆ‘π‘—,π‘˜πžπ£π€, βˆ‘π‘—πžπ£π€(π‘˜βˆˆβ„•), and βˆ‘π‘˜πžπ£π€, (π‘—βˆˆβ„•) by 𝐞, 𝐞𝐀 and 𝐞𝐣, respectively.

Let 𝐸 be the space of double sequences converging with respect to a convergence notion 𝜈, 𝐹 a double sequence space, and 𝐴=(π‘Žπ‘šπ‘›π‘—π‘˜) a 4-dimensional matrix of scalars. Define the set𝐹𝐴(𝜈)ξƒ―[]∢=π‘₯∈Ω∣𝐴π‘₯π‘šπ‘›ξ“βˆΆ=𝜈-𝑗,π‘˜π‘Žπ‘šπ‘›π‘—π‘˜π‘₯π‘—π‘˜ξ€·[]existsand𝐴π‘₯∢=𝐴π‘₯π‘šπ‘›ξ€Έπ‘š,𝑛.∈𝐹(1.4) Then we say that 𝐴 maps the space 𝐸 into the space 𝐹 if πΈβŠ‚πΉπ΄(𝜈) and denote by (𝐸,𝐹) the set of all 4-dimensional matrices 𝐴 which map 𝐸 into 𝐹.

We say that a 4-dimensional matrix 𝐴 is π’žπœˆ-conservative if π’žπœˆβŠ‚π’ž(𝜈)𝜈𝐴, and π’žπœˆ-regular if in addition𝜈-lim𝐴π‘₯∢=𝜈-limπ‘š,𝑛[]𝐴π‘₯π‘šπ‘›=𝜈-limπ‘š,𝑛π‘₯π‘šπ‘›ξ€·π‘₯βˆˆπ’žπœˆξ€Έ,(1.5) whereπ’ž(𝜈)πœˆπ΄ξƒ―[]∢=π‘₯∈Ω∣𝐴π‘₯π‘šπ‘›ξ“βˆΆ=𝜈-𝑗,π‘˜π‘Žπ‘šπ‘›π‘—π‘˜π‘₯π‘—π‘˜ξ€·[]existsand𝐴π‘₯∢=𝐴π‘₯π‘šπ‘›ξ€Έπ‘š,π‘›βˆˆπ’žπœˆξƒ°.(1.6)

Matrix transformations for double sequences are considered by various authors, namely, [3–5].

Let 𝜎 be a one-to-one mapping from the set β„•0={0,1,2,….} into itself. A continuous linear functional πœ‘ on π‘™βˆž is said to be an invariant mean or a 𝜎-mean (see [6, 7]) if and only if (i) πœ‘(π‘₯)β‰₯0 when the sequence π‘₯=(π‘₯π‘˜) has π‘₯π‘˜β‰₯0 for all π‘˜, (ii) πœ‘(𝑒)=1, where 𝑒=(1,1,1,…), and (iii) πœ‘(π‘₯)=πœ‘(π‘₯𝜎(π‘˜)) for all π‘₯βˆˆπ‘™βˆž.

We say that a sequence π‘₯=(π‘₯π‘˜) is 𝜎-convergent to the limit 𝐿 if πœ‘(π‘₯)=𝐿 for all 𝜎-means πœ‘. We denote by π‘‰πœŽ the set of all 𝜎-convergent sequences π‘₯=(π‘₯π‘˜). Clearly π‘βŠ‚π‘‰πœŽ. Note that a 𝜎-mean extends the limit functional on 𝑐 in the sense that πœ‘(π‘₯)=limπ‘₯ for all π‘₯βˆˆπ‘ if and only if 𝜎 has no finite orbits, that is to say, if and only if πœŽπ‘˜(𝑛)≠𝑛, for all 𝑛β‰₯0,π‘˜β‰₯1 (see [8]).

Recently, the concept of invariant mean for double sequences was defined in [9].

Let 𝜎 be a one-to-one mapping from the set β„• of natural numbers into itself. A continuous linear functional πœ‘2 on ℳ𝑒 is said to be an invariant mean or a 𝜎-mean if and only if (i) πœ‘2(π‘₯)β‰₯0 if π‘₯β‰₯0 (i.e., π‘₯π‘—π‘˜β‰₯0 for all 𝑗,π‘˜); (ii) πœ‘2(𝐸)=1, where 𝐸=(π‘’π‘—π‘˜), π‘’π‘—π‘˜=1 for all 𝑗,π‘˜, and (iii) πœ‘2(π‘₯)=πœ‘2((π‘₯𝜎(𝑗),𝜎(π‘˜)))=πœ‘2((π‘₯𝜎(𝑗),π‘˜))=πœ‘2((π‘₯𝑗,𝜎(π‘˜))).

If 𝜎(𝑛)=𝑛+1 then 𝜎-mean is reduced to the Banach limit for double sequences [10].

The idea of 𝜎-convergence for double sequences has recently been introduced in [11] and further studied in [9, 12–16].

A double sequence π‘₯=(π‘₯π‘—π‘˜) of real numbers is said to be 𝜎-convergent to a number 𝐿 if and only if π‘₯βˆˆπ’±πœŽ, whereπ’±πœŽ=ξ‚»π‘₯βˆˆβ„³π‘’βˆΆlim𝑝,π‘žβ†’βˆžπœπ‘π‘žπ‘ π‘‘ξ‚Ό,𝜏(π‘₯)=𝐿uniformlyin𝑠,𝑑;𝐿=πœŽβˆ’limπ‘₯π‘π‘žπ‘ π‘‘βˆΆ=πœπ‘π‘žπ‘ π‘‘(1π‘₯)=(𝑝+1)(π‘ž+1)π‘ξ“π‘žπ‘—=0ξ“π‘˜=0π‘₯πœŽπ‘—(𝑠),πœŽπ‘˜(𝑑).𝜏0π‘žπ‘ π‘‘βˆΆ=𝜏0π‘žπ‘ π‘‘1(π‘₯)=(π‘ž+1)π‘žξ“π‘˜=0π‘₯𝑠,πœŽπ‘˜(𝑑),πœπ‘0π‘ π‘‘βˆΆ=πœπ‘0𝑠𝑑1(π‘₯)=(𝑝+1)π‘βˆ‘π‘—=0π‘₯πœŽπ‘—(𝑠),𝑑,(1.7)𝜏0,0,𝑠,𝑑=π‘₯𝑠𝑑 and πœβˆ’1,π‘ž,𝑠,𝑑=πœπ‘,βˆ’1,𝑠,𝑑=πœβˆ’1,βˆ’1,𝑠,𝑑=0.

Note that π’žπ‘π‘βŠ‚π’±πœŽβŠ‚β„³π‘’.

Throughout this paper limit of a double sequence means 𝑏𝑝-limit.

For 𝜎(𝑛)=𝑛+1, the set π’±πœŽ is reduced to the set 𝑓2 of almost convergent double sequences [17]. A double sequence π‘₯=(π‘₯π‘—π‘˜) of real numbers is said to be almost convergent to a number 𝐿 if and only iflim𝑝,π‘žβ†’βˆž1(𝑝+1)(π‘ž+1)π‘ξ“π‘žπ‘—=0ξ“π‘˜=0π‘₯𝑗+𝑠,π‘˜+𝑑=𝐿uniformlyin𝑠,𝑑.(1.8) The concept of almost convergence for single sequences was introduced by Lorentz [18].

Remark 1.1. In view of the following example, it may be remarked that this does not exclude the possibility that every boundedly convergent double sequence might have a uniquely determined 𝜎-mean not necessarily equal to its 𝑏𝑝-limit.
For example, let 𝜎(𝑛)=0 for all 𝑛. Then it is easily seen that any bounded double sequence (and hence, in particular, any boundedly convergent double sequence) has 𝜎-mean π‘₯00.
In this paper we examine some properties of the invariant mean and define the concepts of absolute 𝜎-convergence and strong 𝜎-convergence for double sequences analogous to the case of single sequences [8, 19]. We further define the absolute invariant mean through which the space of absolutely 𝜎-convergent double sequences is characterized.

2. Strong and Absolute 𝜎-Convergence

In this section we define the concepts of strong 𝜎-convergence and absolute 𝜎-convergence for double sequences. These concepts for single sequences were studied in [8, 19–21].

Remark 2.1. In [9], it was shown that the sublinear functional 𝑉 defined on ℳ𝑒 dominates and generates the 𝜎-means, where π‘‰βˆΆβ„³π‘’β†’β„ is defined by 𝑉(π‘₯)=inf𝑝𝑝=π‘—π‘˜ξ€Έβˆˆπ’±0𝜎limsup𝑗,π‘˜ξ€·π‘₯π‘—π‘˜+π‘π‘—π‘˜ξ€Έ.(2.1)
Now we investigate the sublinear functional which generates the space [π’±πœŽ] of strongly 𝜎-convergent double sequences defined in [22] as ξ€Ίπ’±πœŽξ€»=ξƒ―ξ€·π‘₯π‘₯=π‘—π‘˜ξ€Έβˆˆβ„³π‘’βˆΆlim𝑝,π‘žβ†’βˆž1(𝑝+1)(π‘ž+1)π‘ξ“π‘žπ‘—=0ξ“π‘˜=0||π‘₯πœŽπ‘—(𝑠),πœŽπ‘˜(𝑑)||ξƒ°βˆ’πΏ=0,uniformlyin𝑠,𝑑.(2.2)

Definition 2.2. We define Ξ¨βˆΆβ„³π‘’β†’β„ by Ξ¨(π‘₯)=limsup𝑝,π‘žsup𝑠,𝑑1(𝑝+1)(π‘ž+1)π‘ξ“π‘žπ‘—=0ξ“π‘˜=0||π‘₯πœŽπ‘—(𝑠),πœŽπ‘˜(𝑑)||.(2.3)
Let {ℳ𝑒,Ξ¨} denote the set of all linear functionals Ξ¦ on ℳ𝑒 such that Ξ¦(π‘₯)≀Ψ(π‘₯) for all π‘₯=(π‘₯π‘—π‘˜)βˆˆβ„³π‘’. By Hahn-Banach Theorem, the set {ℳ𝑒,Ξ¨} is nonempty.
If there exists πΏβˆˆβ„ such that Ξ¦ξ€½β„³(π‘₯βˆ’πΏπž)=0βˆ€Ξ¦βˆˆπ‘’ξ€Ύ,Ξ¨,(βˆ—) then we say that π‘₯ is {ℳ𝑒,Ξ¨}-convergent to 𝐿 and in this case we write {ℳ𝑒,Ξ¨}-limπ‘₯=𝐿.

We are now ready to prove the following result.

Theorem 2.3. [π’±πœŽ] is the set of all {ℳ𝑒,Ξ¨}-convergent sequences.

Proof. Let π‘₯∈[π’±πœŽ]. Then for each πœ–>0, there exist 𝑝0,π‘ž0 such that for 𝑝>𝑝0,π‘ž>π‘ž0 and all 𝑠,𝑑, 1(𝑝+1)(π‘ž+1)π‘ξ“π‘žπ‘—=0ξ“π‘˜=0||π‘₯πœŽπ‘—(𝑠),πœŽπ‘˜(𝑑)||βˆ’πΏ<πœ–,(2.4) and this implies that Ξ¨(π‘₯βˆ’πΏπž)β‰€πœ–. In a similar manner, we can prove that Ξ¨(πΏπžβˆ’π‘₯)β‰€πœ–. Hence |Ξ¦(π‘₯βˆ’πΏπž)|≀Ψ(π‘₯βˆ’πΏπž)β‰€πœ– for all Φ∈{ℳ𝑒,Ξ¨}. Therefore Ξ¦(π‘₯βˆ’πΏπž)=0for all Φ∈{ℳ𝑒,Ξ¨} and this implies that by (2.6) π‘₯∈[π’±πœŽ] implies that π‘₯ is {ℳ𝑒,Ξ¨}-convergent.
Conversely, suppose that π‘₯ is {ℳ𝑒,Ξ¨}-convergent, that is, Ξ¦ξ€½β„³(π‘₯βˆ’πΏπž)=0βˆ€Ξ¦βˆˆπ‘’ξ€Ύ,Ξ¨.(2.5)
Since Ξ¨ is sublinear functional on ℳ𝑒, by Hahn-Banach Theorem, there exists Ξ¦0∈{ℳ𝑒,Ξ¨} such that Ξ¦0(π‘₯βˆ’πΏπž)=Ξ¨(π‘₯βˆ’πΏπž). Hence Ξ¨(π‘₯βˆ’πΏπž)=0; since Ξ¨(π‘₯)=Ξ¨(βˆ’π‘₯), it follows that π‘₯∈[π’±πœŽ]. This completes the proof of the theorem.

Now we define the concept of absolute 𝜎-convergence for double sequences.

Putπœ™π‘π‘žπ‘ π‘‘(π‘₯)=πœπ‘π‘žπ‘ π‘‘(π‘₯)βˆ’πœπ‘βˆ’1,π‘ž,𝑠,𝑑(π‘₯)βˆ’πœπ‘,π‘žβˆ’1,𝑠,𝑑(π‘₯)+πœπ‘βˆ’1,π‘žβˆ’1,𝑠,𝑑(π‘₯).(2.6) Thus simplifying further, we getπœ™π‘π‘žπ‘ π‘‘1(π‘₯)=𝑝(𝑝+1)π‘ξ“π‘š=1π‘šξƒ¬1π‘ž(π‘ž+1)π‘žξ“π‘›=1𝑛π‘₯πœŽπ‘š(𝑠),πœŽπ‘›(𝑑)βˆ’π‘₯πœŽπ‘š(𝑠),πœŽπ‘›βˆ’1(𝑑)ξ€Έξƒ­=1×𝑝(𝑝+1)π‘ž(π‘ž+1)π‘ξ“π‘žπ‘š=1𝑛=1ξ€Ίπ‘₯π‘šπ‘›πœŽπ‘š(𝑠),πœŽπ‘›(𝑑)βˆ’π‘₯πœŽπ‘šβˆ’1(𝑠),πœŽπ‘›(𝑑)βˆ’π‘₯πœŽπ‘š(𝑠),πœŽπ‘›βˆ’1(𝑑)+π‘₯πœŽπ‘šβˆ’1(𝑠),πœŽπ‘›βˆ’1(𝑑)ξ€».(2.7) Now we writeπœ™π‘π‘žπ‘ π‘‘βŽ§βŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺ⎩1(π‘₯)=𝑝×(𝑝+1)π‘ž(π‘ž+1)π‘ξ“π‘žπ‘š=1𝑛=1ξ€Ίπ‘₯π‘šπ‘›πœŽπ‘š(𝑠),πœŽπ‘›(𝑑)βˆ’π‘₯πœŽπ‘šβˆ’1(𝑠),πœŽπ‘›(𝑑)βˆ’π‘₯πœŽπ‘š(𝑠),πœŽπ‘›βˆ’1(𝑑)+π‘₯πœŽπ‘šβˆ’1(𝑠),πœŽπ‘›βˆ’1(𝑑)ξ€»1,𝑝,π‘žβ‰₯1,π‘ž(π‘ž+1)π‘žξ“π‘›=1𝑛π‘₯𝑠,πœŽπ‘›(𝑑)βˆ’π‘₯𝑠,πœŽπ‘›βˆ’1(𝑑)ξ€»1,𝑝=0,π‘žβ‰₯1,𝑝(𝑝+1)𝑝𝑛=1π‘šξ€Ίπ‘₯πœŽπ‘š(𝑠),π‘‘βˆ’π‘₯πœŽπ‘š(𝑠),𝑑,𝑝β‰₯1,π‘ž=0,(2.8) and πœ™00𝑠𝑑(π‘₯)=π‘₯𝑠𝑑.

In [9], the following was defined.

Definition 2.4. A double sequence π‘₯=(π‘₯π‘—π‘˜)βˆˆβ„³π‘’ is said to be absolutelyΟƒ-almost convergent if and only if βˆžξ“βˆžπ‘=0ξ“π‘ž=0||πœ™π‘π‘žπ‘ π‘‘(||π‘₯)convergesuniformlyin𝑠,𝑑.(2.9)
By π’²πœŽ, we denote the space of all absolutely 𝜎-almost convergent double sequences.

Now we define the following.

Definition 2.5. A double sequence π‘₯=(π‘₯π‘—π‘˜)βˆˆβ„³π‘’ is said to be absolutely Οƒ-convergent if and only if(i)βˆ‘βˆžπ‘=0βˆ‘βˆžπ‘ž=0|πœ™π‘π‘žπ‘ π‘‘(π‘₯)| converges uniformly in 𝑠,𝑑;(ii)lim𝑝,π‘žβ†’βˆžπœπ‘π‘žπ‘ π‘‘(π‘₯), which must exist, should take the same value for all 𝑠,𝑑.
By β„¬π’±πœŽ, we denote the space of all absolutely 𝜎-convergent double sequences. It is easy to prove that both π’²πœŽ and β„¬π’±πœŽ are Banach spaces with the norm β€–π‘₯β€–=supβˆžπ‘ ,π‘‘ξ“βˆžπ‘=0ξ“π‘ž=0||πœ™π‘π‘žπ‘ π‘‘(||π‘₯).(2.10)

Note that β„¬π’±πœŽβŠ‚π’²πœŽβŠ‚π’±πœŽ.

Remark 2.6. It is easy to see that the assertion (i) implies that (πœπ‘π‘žπ‘ π‘‘(π‘₯)) (as a double sequence in 𝑝,π‘ž) converges uniformly in 𝑠,𝑑, but it may converge to a different limit for different values of 𝑠,𝑑. This point did not arise in Banach limit case in which 𝜎(𝑛)=𝑛+1. In this case if we assume only that lim𝑝,π‘žβ†’βˆžπœπ‘π‘žπ‘ π‘‘(π‘₯)=β„“ for some value of 𝑠,𝑑; then we must have lim𝑝,π‘žβ†’βˆžπœπ‘π‘žπ‘ π‘‘(π‘₯)=β„“ for any other 𝑠,𝑑 (but not necessarily uniformly in 𝑠,𝑑). So if, as a special case, we assume uniform convergence, the value to πœπ‘π‘žπ‘ π‘‘(π‘₯) converges must be same for all 𝑠,𝑑. This need not be in the general case. For example, consider 𝜎(𝑛)=𝑛+2. Define the sequence π‘₯=(π‘₯π‘—π‘˜) by π‘₯π‘—π‘˜=ξ‚»1,if𝑗isodd,βˆ€π‘˜,0,if𝑗iseven,βˆ€π‘˜.(2.11) Then for all 𝑝,π‘žβ‰₯0πœπ‘π‘žπ‘ π‘‘(⎧βŽͺ⎨βŽͺ⎩π‘₯)=1,if𝑠isodd,βˆ€π‘‘,0,if𝑠iseven,βˆ€π‘‘,0,otherwise,(2.12) so that πœ™π‘π‘žπ‘ π‘‘(π‘₯)=0 for all 𝑝,π‘žβ‰₯1 (in particular, πœ™1111(π‘₯)=π‘₯𝜎(1),𝜎(1)βˆ’π‘₯11=π‘₯3,3βˆ’π‘₯11=1βˆ’1=0, since 𝜎(1)=1+2=3). Thus (i) certainly holds, but the value of lim𝑝,π‘žβ†’βˆžπœπ‘π‘žπ‘ π‘‘(π‘₯) is 1 when 𝑠 is odd and 0 when 𝑠 is even (for all 𝑑). Moreover, it shows that the inclusion β„¬π’±πœŽβŠ‚π’²πœŽ is proper.

3. Absolute Invariant Mean

Remark 3.1. It may be remarked that we have a class of linear continuous functionals πœ‘2 on ℳ𝑒 (which we call the set of invariant means) such that πœ‘2 is uniquely determined if and only if π‘₯βˆˆπ’±πœŽ, that is, the largest set which determines πœ‘2 uniquely is π’±πœŽ. Now we are going to deal with the similar situation which prevails for β„¬π’±πœŽ.

As an immediate consequence, we have the following.

Theorem 3.2. There does not exist a class of continuous linear functionals πœ‘2 on ℳ𝑒 such that πœ‘2 is uniquely determined if and only if π‘₯βˆˆβ„¬π’±πœŽ.

Proof. We first note that β„¬π’±πœŽ is not closed in ℳ𝑒 (which follows from the case 𝜎(𝑛)=𝑛+1 for single sequences which is proved in [23]). Given the value of πœ‘2(π‘₯) for π‘₯βˆˆβ„¬π’±πœŽ, its value for π‘₯∈cl(β„¬π’±πœŽ) is determined by continuity. So if πœ‘2(π‘₯) is unique for π‘₯βˆˆβ„¬π’±πœŽ, it must be unique in the set cl(β„¬π’±πœŽ), which is larger than β„¬π’±πœŽ.

Remark 3.3. As in Remark 2.1, it is easy to see that the sublinear functional πœ†(π‘₯)=limsup𝑝,π‘žsup𝑠,π‘‘πœπ‘π‘žπ‘ π‘‘(π‘₯)(3.1) both dominates and generates the functional πœ‘2 which is a 𝜎-mean if and only if βˆ’πœ†(βˆ’π‘₯)β‰€πœ‘2(π‘₯)β‰€πœ†(π‘₯).(3.2) It follows from (3.2) that πœ‘2 is unique 𝜎-mean if and only if π’±πœŽ=ξ€½π‘₯βˆˆβ„³π‘’ξ€ΎβˆΆπœ†(π‘₯)=βˆ’πœ†(βˆ’π‘₯).(3.3)
In the same vein, we seek a characterization of a class of linear functionals πœ“2 on ℳ𝑒 to define absolute invariant mean in terms of a suitable sublinear functional 𝑄 on ℳ𝑒.

Definition 3.4. A linear functional πœ“2 on ℳ𝑒 is an absolute invariant mean (π’œβ„β„³) if and only if βˆ’π‘„(βˆ’π‘₯)β‰€πœ“2(π‘₯)≀𝑄(π‘₯) and is unique π’œβ„β„³ if and only if ξ€·clβ„¬π’±πœŽξ€Έ=ξ€½π‘₯βˆˆβ„³π‘’ξ€ΎβˆΆπ‘„(π‘₯)=βˆ’π‘„(βˆ’π‘₯),(3.4) where 𝑄(π‘₯)=limsup𝑝,π‘žsupβˆžπ‘ ,π‘‘ξ“βˆžπ‘–=𝑝𝑗=π‘ž||πœ™π‘–π‘—π‘ π‘‘(||π‘₯)<∞.(3.5)

We have the following result.

Theorem 3.5. One has β„¬π’±πœŽ=ξ€½π‘₯βˆˆβ„³π‘’ξ€ΎβˆΆπ‘„(π‘₯)=0.(3.6)

Proof. Since 𝑄 is a sublinear functional on ℳ𝑒, it follows from Hahn-Banach Theorem that there exists a continuous linear functional πœ‡ on ℳ𝑒 such that πœ‡(π‘₯)≀𝑄(π‘₯)βˆ€π‘₯βˆˆβ„³π‘’,(3.7) and this limit is unique if and only if 𝑄(π‘₯)=βˆ’π‘„(βˆ’π‘₯)=βˆ’π‘„(π‘₯), that is, if and only if 𝑄(π‘₯)=0 for all π‘₯βˆˆβ„³π‘’. That is, if and only if limβˆžπ‘,π‘žξ“βˆžπ‘–=𝑝𝑗=π‘ž||πœ™π‘–π‘—π‘ π‘‘(||π‘₯)=0uniformlyin𝑠,𝑑,(3.8) that is, if and only if π‘₯βˆˆβ„¬π’±πœŽ.

Acknowledgments

The authors would like to thank the Deanship of Scientific Research at King Abdulaziz University for its financial support under Grant no. 99-130-1432. The authors are also thankful to the referee for his/her constructive comments which helped to improve the present paper.

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