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

Invariant and Absolute Invariant Means of Double Sequences

Abdullah Alotaibi,1Β M. Mursaleen,2Β and M. A. Alghamdi1

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‘₯ 𝑗 π‘˜ ) of real or complex numbers is said to be bounded if β€– π‘₯ β€– ∞ = s u p 𝑗 , π‘˜ | | π‘₯ 𝑗 π‘˜ | | < ∞ . ( 1 . 1 ) The space of all bounded double sequences is denoted by β„³ 𝑒 .

A double sequence π‘₯ = ( π‘₯ 𝑗 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‘₯ 𝑗 π‘˜ βˆ’ 𝐿 | < πœ€ whenever 𝑗 , π‘˜ > 𝑁 . In this case 𝐿 is called the 𝑝 -limit of π‘₯ . If in addition π‘₯ ∈ β„³ 𝑒 , then π‘₯ is said to be boundedly convergent to L in Pringsheim’s sense (shortly, bp-convergent to L).

A double sequence π‘₯ = ( π‘₯ 𝑗 π‘˜ ) is said to converge regularly to L (shortly, r-convergent to L) if π‘₯ is 𝑝 -convergent and the limits π‘₯ 𝑗 ∢ = l i m π‘˜ π‘₯ 𝑗 π‘˜ ( 𝑗 ∈ β„• ) and π‘₯ π‘˜ ∢ = l i m 𝑗 π‘₯ 𝑗 π‘˜ ( π‘˜ ∈ β„• ) exist. Note that in this case the limits l i m 𝑗 l i m π‘˜ π‘₯ 𝑗 π‘˜ and l i m π‘˜ l i m 𝑗 π‘₯ 𝑗 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’ž 𝜈 and the limit of a 𝜈 -convergent double sequence π‘₯ by 𝜈 - l i m 𝑗 , π‘˜ π‘₯ 𝑗 π‘˜ , where 𝜈 ∈ { 𝑝 , 𝑏 𝑝 , 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 contain ξ€½ 𝐞 s p a n 𝐣 𝐀 ξ€Ύ , ∣ 𝑗 , π‘˜ ∈ β„• ( 1 . 2 ) where 𝐞 𝐣 𝐀 𝐒 π₯ = ξ‚» 1 , i f ( 𝑗 , π‘˜ ) = ( 𝑖 , β„“ ) , 0 , o t h e r w i s e . ( 1 . 3 )

We denote the pointwise sums βˆ‘ 𝑗 , π‘˜ 𝐞 𝐣 𝐀 , βˆ‘ 𝑗 𝐞 𝐣 𝐀 ( π‘˜ ∈ β„• ) , and βˆ‘ π‘˜ 𝐞 𝐣 𝐀 , ( 𝑗 ∈ β„• ) by 𝐞 , 𝐞 𝐀 and 𝐞 𝐣 , 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‘Ž π‘š 𝑛 𝑗 π‘˜ ) a 4-dimensional matrix of scalars. Define the set 𝐹 𝐴 ( 𝜈 ) ξƒ― [ ] ∢ = π‘₯ ∈ Ξ© ∣ 𝐴 π‘₯ π‘š 𝑛  ∢ = 𝜈 - 𝑗 , π‘˜ π‘Ž π‘š 𝑛 𝑗 π‘˜ π‘₯ 𝑗 π‘˜ ξ€· [ ] e x i s t s a n d 𝐴 π‘₯ ∢ = 𝐴 π‘₯ π‘š 𝑛 ξ€Έ π‘š , 𝑛 ξƒ° . ∈ 𝐹 ( 1 . 4 ) Then we say that 𝐴 maps the space 𝐸 into the space 𝐹 if 𝐸 βŠ‚ 𝐹 𝐴 ( 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’ž 𝜈 -conservative if π’ž 𝜈 βŠ‚ π’ž ( 𝜈 ) 𝜈 𝐴 , and π’ž 𝜈 -regular if in addition 𝜈 - l i m 𝐴 π‘₯ ∢ = 𝜈 - l i m π‘š , 𝑛 [ ] 𝐴 π‘₯ π‘š 𝑛 = 𝜈 - l i m π‘š , 𝑛 π‘₯ π‘š 𝑛 ξ€· π‘₯ ∈ π’ž 𝜈 ξ€Έ , ( 1 . 5 ) where π’ž ( 𝜈 ) 𝜈 𝐴 ξƒ― [ ] ∢ = π‘₯ ∈ Ξ© ∣ 𝐴 π‘₯ π‘š 𝑛  ∢ = 𝜈 - 𝑗 , π‘˜ π‘Ž π‘š 𝑛 𝑗 π‘˜ π‘₯ 𝑗 π‘˜ ξ€· [ ] e x i s t s a n d 𝐴 π‘₯ ∢ = 𝐴 π‘₯ π‘š 𝑛 ξ€Έ π‘š , 𝑛 ∈ π’ž 𝜈 ξƒ° . ( 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 𝑙 ∞ 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‘₯ ∈ 𝑙 ∞ .

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 𝑉 𝜎 the set of all 𝜎 -convergent sequences π‘₯ = ( π‘₯ π‘˜ ) . Clearly 𝑐 βŠ‚ 𝑉 𝜎 . Note that a 𝜎 -mean extends the limit functional on 𝑐 in the sense that πœ‘ ( π‘₯ ) = l i m π‘₯ for all π‘₯ ∈ 𝑐 if and only if 𝜎 has no finite orbits, that is to say, if and only if 𝜎 π‘˜ ( 𝑛 ) β‰  𝑛 , for all 𝑛 β‰₯ 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 β„³ 𝑒 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‘₯ 𝑗 π‘˜ β‰₯ 0 for all 𝑗 , π‘˜ ); (ii) πœ‘ 2 ( 𝐸 ) = 1 , where 𝐸 = ( 𝑒 𝑗 π‘˜ ) , 𝑒 𝑗 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‘₯ 𝑗 π‘˜ ) of real numbers is said to be 𝜎 -convergent to a number 𝐿 if and only if π‘₯ ∈ 𝒱 𝜎 , where 𝒱 𝜎 = ξ‚» π‘₯ ∈ β„³ 𝑒 ∢ l i m 𝑝 , π‘ž β†’ ∞ 𝜏 𝑝 π‘ž 𝑠 𝑑 ξ‚Ό , 𝜏 ( π‘₯ ) = 𝐿 u n i f o r m l y i n 𝑠 , 𝑑 ; 𝐿 = 𝜎 βˆ’ l i m π‘₯ 𝑝 π‘ž 𝑠 𝑑 ∢ = 𝜏 𝑝 π‘ž 𝑠 𝑑 ( 1 π‘₯ ) = ( 𝑝 + 1 ) ( π‘ž + 1 ) 𝑝  π‘ž 𝑗 = 0  π‘˜ = 0 π‘₯ 𝜎 𝑗 ( 𝑠 ) , 𝜎 π‘˜ ( 𝑑 ) . 𝜏 0 π‘ž 𝑠 𝑑 ∢ = 𝜏 0 π‘ž 𝑠 𝑑 1 ( π‘₯ ) = ( π‘ž + 1 ) π‘ž  π‘˜ = 0 π‘₯ 𝑠 , 𝜎 π‘˜ ( 𝑑 ) , 𝜏 𝑝 0 𝑠 𝑑 ∢ = 𝜏 𝑝 0 𝑠 𝑑 1 ( π‘₯ ) = ( 𝑝 + 1 ) 𝑝 βˆ‘ 𝑗 = 0 π‘₯ 𝜎 𝑗 ( 𝑠 ) , 𝑑 , ( 1 . 7 ) 𝜏 0 , 0 , 𝑠 , 𝑑 = π‘₯ 𝑠 𝑑 and 𝜏 βˆ’ 1 , π‘ž , 𝑠 , 𝑑 = 𝜏 𝑝 , βˆ’ 1 , 𝑠 , 𝑑 = 𝜏 βˆ’ 1 , βˆ’ 1 , 𝑠 , 𝑑 = 0 .

Note that π’ž 𝑏 𝑝 βŠ‚ 𝒱 𝜎 βŠ‚ β„³ 𝑒 .

Throughout this paper limit of a double sequence means 𝑏 𝑝 -limit.

For 𝜎 ( 𝑛 ) = 𝑛 + 1 , the set 𝒱 𝜎 is reduced to the set 𝑓 2 of almost convergent double sequences [17]. A double sequence π‘₯ = ( π‘₯ 𝑗 π‘˜ ) of real numbers is said to be almost convergent to a number 𝐿 if and only if l i m 𝑝 , π‘ž β†’ ∞ 1 ( 𝑝 + 1 ) ( π‘ž + 1 ) 𝑝  π‘ž 𝑗 = 0  π‘˜ = 0 π‘₯ 𝑗 + 𝑠 , π‘˜ + 𝑑 = 𝐿 u n i f o r m l y i n 𝑠 , 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 𝑏 𝑝 -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‘₯ 0 0 .
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 β„³ 𝑒 dominates and generates the 𝜎 -means, where 𝑉 ∢ β„³ 𝑒 β†’ ℝ is defined by 𝑉 ( π‘₯ ) = i n f ξ€· 𝑝 𝑝 = 𝑗 π‘˜ ξ€Έ ∈ 𝒱 0 𝜎 l i m s u p 𝑗 , π‘˜ ξ€· π‘₯ 𝑗 π‘˜ + 𝑝 𝑗 π‘˜ ξ€Έ . ( 2 . 1 )
Now we investigate the sublinear functional which generates the space [ 𝒱 𝜎 ] of strongly 𝜎 -convergent double sequences defined in [22] as ξ€Ί 𝒱 𝜎 ξ€» = ξƒ― ξ€· π‘₯ π‘₯ = 𝑗 π‘˜ ξ€Έ ∈ β„³ 𝑒 ∢ l i m 𝑝 , π‘ž β†’ ∞ 1 ( 𝑝 + 1 ) ( π‘ž + 1 ) 𝑝  π‘ž 𝑗 = 0  π‘˜ = 0 | | π‘₯ 𝜎 𝑗 ( 𝑠 ) , 𝜎 π‘˜ ( 𝑑 ) | | ξƒ° βˆ’ 𝐿 = 0 , u n i f o r m l y i n 𝑠 , 𝑑 . ( 2 . 2 )

Definition 2.2. We define Ξ¨ ∢ β„³ 𝑒 β†’ ℝ by Ξ¨ ( π‘₯ ) = l i m s u p 𝑝 , π‘ž s u p 𝑠 , 𝑑 1 ( 𝑝 + 1 ) ( π‘ž + 1 ) 𝑝  π‘ž 𝑗 = 0  π‘˜ = 0 | | π‘₯ 𝜎 𝑗 ( 𝑠 ) , 𝜎 π‘˜ ( 𝑑 ) | | . ( 2 . 3 )
Let { β„³ 𝑒 , Ξ¨ } denote the set of all linear functionals Ξ¦ on β„³ 𝑒 such that Ξ¦ ( π‘₯ ) ≀ Ξ¨ ( π‘₯ ) for all π‘₯ = ( π‘₯ 𝑗 π‘˜ ) ∈ β„³ 𝑒 . By Hahn-Banach Theorem, the set { β„³ 𝑒 , Ξ¨ } is nonempty.
If there exists 𝐿 ∈ ℝ such that Ξ¦ ξ€½ β„³ ( π‘₯ βˆ’ 𝐿 𝐞 ) = 0 βˆ€ Ξ¦ ∈ 𝑒 ξ€Ύ , Ξ¨ , ( βˆ— ) then we say that π‘₯ is { β„³ 𝑒 , Ξ¨ } -convergent to 𝐿 and in this case we write { β„³ 𝑒 , Ξ¨ } - l i m π‘₯ = 𝐿 .

We are now ready to prove the following result.

Theorem 2.3. [ 𝒱 𝜎 ] is the set of all { β„³ 𝑒 , Ξ¨ } -convergent sequences.

Proof. Let π‘₯ ∈ [ 𝒱 𝜎 ] . Then for each πœ– > 0 , there exist 𝑝 0 , π‘ž 0 such that for 𝑝 > 𝑝 0 , π‘ž > π‘ž 0 and all 𝑠 , 𝑑 , 1 ( 𝑝 + 1 ) ( π‘ž + 1 ) 𝑝  π‘ž 𝑗 = 0  π‘˜ = 0 | | π‘₯ 𝜎 𝑗 ( 𝑠 ) , 𝜎 π‘˜ ( 𝑑 ) | | βˆ’ 𝐿 < πœ– , ( 2 . 4 ) and this implies that Ξ¨ ( π‘₯ βˆ’ 𝐿 𝐞 ) ≀ πœ– . In a similar manner, we can prove that Ξ¨ ( 𝐿 𝐞 βˆ’ π‘₯ ) ≀ πœ– . Hence | Ξ¦ ( π‘₯ βˆ’ 𝐿 𝐞 ) | ≀ Ξ¨ ( π‘₯ βˆ’ 𝐿 𝐞 ) ≀ πœ– for all Ξ¦ ∈ { β„³ 𝑒 , Ξ¨ } . Therefore Ξ¦ ( π‘₯ βˆ’ 𝐿 𝐞 ) = 0 for all Ξ¦ ∈ { β„³ 𝑒 , Ξ¨ } and this implies that by (2.6) π‘₯ ∈ [ 𝒱 𝜎 ] implies that π‘₯ is { β„³ 𝑒 , Ξ¨ } -convergent.
Conversely, suppose that π‘₯ is { β„³ 𝑒 , Ξ¨ } -convergent, that is, Ξ¦ ξ€½ β„³ ( π‘₯ βˆ’ 𝐿 𝐞 ) = 0 βˆ€ Ξ¦ ∈ 𝑒 ξ€Ύ , Ξ¨ . ( 2 . 5 )
Since Ξ¨ is sublinear functional on β„³ 𝑒 , by Hahn-Banach Theorem, there exists Ξ¦ 0 ∈ { β„³ 𝑒 , Ξ¨ } such that Ξ¦ 0 ( π‘₯ βˆ’ 𝐿 𝐞 ) = Ξ¨ ( π‘₯ βˆ’ 𝐿 𝐞 ) . Hence Ξ¨ ( π‘₯ βˆ’ 𝐿 𝐞 ) = 0 ; since Ξ¨ ( π‘₯ ) = Ξ¨ ( βˆ’ π‘₯ ) , it follows that π‘₯ ∈ [ 𝒱 𝜎 ] . This completes the proof of the theorem.

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

Put πœ™ 𝑝 π‘ž 𝑠 𝑑 ( π‘₯ ) = 𝜏 𝑝 π‘ž 𝑠 𝑑 ( π‘₯ ) βˆ’ 𝜏 𝑝 βˆ’ 1 , π‘ž , 𝑠 , 𝑑 ( π‘₯ ) βˆ’ 𝜏 𝑝 , π‘ž βˆ’ 1 , 𝑠 , 𝑑 ( π‘₯ ) + 𝜏 𝑝 βˆ’ 1 , π‘ž βˆ’ 1 , 𝑠 , 𝑑 ( π‘₯ ) . ( 2 . 6 ) Thus simplifying further, we get πœ™ 𝑝 π‘ž 𝑠 𝑑 1 ( π‘₯ ) = 𝑝 ( 𝑝 + 1 ) 𝑝  π‘š = 1 π‘š  1 π‘ž ( π‘ž + 1 ) π‘ž  𝑛 = 1 𝑛 ξ€· π‘₯ 𝜎 π‘š ( 𝑠 ) , 𝜎 𝑛 ( 𝑑 ) βˆ’ π‘₯ 𝜎 π‘š ( 𝑠 ) , 𝜎 𝑛 βˆ’ 1 ( 𝑑 ) ξ€Έ ξƒ­ = 1 Γ— 𝑝 ( 𝑝 + 1 ) π‘ž ( π‘ž + 1 ) 𝑝  π‘ž π‘š = 1  𝑛 = 1 ξ€Ί π‘₯ π‘š 𝑛 𝜎 π‘š ( 𝑠 ) , 𝜎 𝑛 ( 𝑑 ) βˆ’ π‘₯ 𝜎 π‘š βˆ’ 1 ( 𝑠 ) , 𝜎 𝑛 ( 𝑑 ) βˆ’ π‘₯ 𝜎 π‘š ( 𝑠 ) , 𝜎 𝑛 βˆ’ 1 ( 𝑑 ) + π‘₯ 𝜎 π‘š βˆ’ 1 ( 𝑠 ) , 𝜎 𝑛 βˆ’ 1 ( 𝑑 ) ξ€» . ( 2 . 7 ) Now we write πœ™ 𝑝 π‘ž 𝑠 𝑑 ⎧ βŽͺ βŽͺ βŽͺ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ βŽͺ βŽͺ βŽͺ ⎩ 1 ( π‘₯ ) = 𝑝 Γ— ( 𝑝 + 1 ) π‘ž ( π‘ž + 1 ) 𝑝  π‘ž π‘š = 1  𝑛 = 1 ξ€Ί π‘₯ π‘š 𝑛 𝜎 π‘š ( 𝑠 ) , 𝜎 𝑛 ( 𝑑 ) βˆ’ π‘₯ 𝜎 π‘š βˆ’ 1 ( 𝑠 ) , 𝜎 𝑛 ( 𝑑 ) βˆ’ π‘₯ 𝜎 π‘š ( 𝑠 ) , 𝜎 𝑛 βˆ’ 1 ( 𝑑 ) + π‘₯ 𝜎 π‘š βˆ’ 1 ( 𝑠 ) , 𝜎 𝑛 βˆ’ 1 ( 𝑑 ) ξ€» 1 , 𝑝 , π‘ž β‰₯ 1 , π‘ž ( π‘ž + 1 ) π‘ž  𝑛 = 1 𝑛 ξ€Ί π‘₯ 𝑠 , 𝜎 𝑛 ( 𝑑 ) βˆ’ π‘₯ 𝑠 , 𝜎 𝑛 βˆ’ 1 ( 𝑑 ) ξ€» 1 , 𝑝 = 0 , π‘ž β‰₯ 1 , 𝑝 ( 𝑝 + 1 ) 𝑝  𝑛 = 1 π‘š ξ€Ί π‘₯ 𝜎 π‘š ( 𝑠 ) , 𝑑 βˆ’ π‘₯ 𝜎 π‘š ( 𝑠 ) , 𝑑 ξ€» , 𝑝 β‰₯ 1 , π‘ž = 0 , ( 2 . 8 ) and πœ™ 0 0 𝑠 𝑑 ( π‘₯ ) = π‘₯ 𝑠 𝑑 .

In [9], the following was defined.

Definition 2.4. A double sequence π‘₯ = ( π‘₯ 𝑗 π‘˜ ) ∈ β„³ 𝑒 is said to be absolutelyσ-almost convergent if and only if ∞  ∞ 𝑝 = 0  π‘ž = 0 | | πœ™ 𝑝 π‘ž 𝑠 𝑑 ( | | π‘₯ ) c o n v e r g e s u n i f o r m l y i n 𝑠 , 𝑑 . ( 2 . 9 )
By 𝒲 𝜎 , 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‘₯ 𝑗 π‘˜ ) ∈ β„³ 𝑒 is said to be absolutely σ-convergent if and only if(i) βˆ‘ ∞ 𝑝 = 0 βˆ‘ ∞ π‘ž = 0 | πœ™ 𝑝 π‘ž 𝑠 𝑑 ( π‘₯ ) | converges uniformly in 𝑠 , 𝑑 ;(ii) l i m 𝑝 , π‘ž β†’ ∞ 𝜏 𝑝 π‘ž 𝑠 𝑑 ( π‘₯ ) , which must exist, should take the same value for all 𝑠 , 𝑑 .
By ℬ 𝒱 𝜎 , we denote the space of all absolutely 𝜎 -convergent double sequences. It is easy to prove that both 𝒲 𝜎 and ℬ 𝒱 𝜎 are Banach spaces with the norm β€– π‘₯ β€– = s u p ∞ 𝑠 , 𝑑  ∞ 𝑝 = 0  π‘ž = 0 | | πœ™ 𝑝 π‘ž 𝑠 𝑑 ( | | π‘₯ ) . ( 2 . 1 0 )

Note that ℬ 𝒱 𝜎 βŠ‚ 𝒲 𝜎 βŠ‚ 𝒱 𝜎 .

Remark 2.6. It is easy to see that the assertion (i) implies that ( 𝜏 𝑝 π‘ž 𝑠 𝑑 ( 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 l i m 𝑝 , π‘ž β†’ ∞ 𝜏 𝑝 π‘ž 𝑠 𝑑 ( π‘₯ ) = β„“ for some value of 𝑠 , 𝑑 ; then we must have l i m 𝑝 , π‘ž β†’ ∞ 𝜏 𝑝 π‘ž 𝑠 𝑑 ( 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 𝜏 𝑝 π‘ž 𝑠 𝑑 ( 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‘₯ 𝑗 π‘˜ ) by π‘₯ 𝑗 π‘˜ = ξ‚» 1 , i f 𝑗 i s o d d , βˆ€ π‘˜ , 0 , i f 𝑗 i s e v e n , βˆ€ π‘˜ . ( 2 . 1 1 ) Then for all 𝑝 , π‘ž β‰₯ 0 𝜏 𝑝 π‘ž 𝑠 𝑑 ( ⎧ βŽͺ ⎨ βŽͺ ⎩ π‘₯ ) = 1 , i f 𝑠 i s o d d , βˆ€ 𝑑 , 0 , i f 𝑠 i s e v e n , βˆ€ 𝑑 , 0 , o t h e r w i s e , ( 2 . 1 2 ) so that πœ™ 𝑝 π‘ž 𝑠 𝑑 ( π‘₯ ) = 0 for all 𝑝 , π‘ž β‰₯ 1 (in particular, πœ™ 1 1 1 1 ( π‘₯ ) = π‘₯ 𝜎 ( 1 ) , 𝜎 ( 1 ) βˆ’ π‘₯ 1 1 = π‘₯ 3 , 3 βˆ’ π‘₯ 1 1 = 1 βˆ’ 1 = 0 , since 𝜎 ( 1 ) = 1 + 2 = 3 ) . Thus (i) certainly holds, but the value of l i m 𝑝 , π‘ž β†’ ∞ 𝜏 𝑝 π‘ž 𝑠 𝑑 ( π‘₯ ) is 1 when 𝑠 is odd and 0 when 𝑠 is even (for all 𝑑 ). Moreover, it shows that the inclusion ℬ 𝒱 𝜎 βŠ‚ 𝒲 𝜎 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 β„³ 𝑒 (which we call the set of invariant means) such that πœ‘ 2 is uniquely determined if and only if π‘₯ ∈ 𝒱 𝜎 , that is, the largest set which determines πœ‘ 2 uniquely is 𝒱 𝜎 . Now we are going to deal with the similar situation which prevails for ℬ 𝒱 𝜎 .

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 β„³ 𝑒 such that πœ‘ 2 is uniquely determined if and only if π‘₯ ∈ ℬ 𝒱 𝜎 .

Proof. We first note that ℬ 𝒱 𝜎 is not closed in β„³ 𝑒 (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‘₯ ∈ ℬ 𝒱 𝜎 , its value for π‘₯ ∈ c l ( ℬ 𝒱 𝜎 ) is determined by continuity. So if πœ‘ 2 ( π‘₯ ) is unique for π‘₯ ∈ ℬ 𝒱 𝜎 , it must be unique in the set c l ( ℬ 𝒱 𝜎 ) , which is larger than ℬ 𝒱 𝜎 .

Remark 3.3. As in Remark 2.1, it is easy to see that the sublinear functional πœ† ( π‘₯ ) = l i m s u p 𝑝 , π‘ž s u p 𝑠 , 𝑑 𝜏 𝑝 π‘ž 𝑠 𝑑 ( 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‘₯ ) ≀ πœ‘ 2 ( π‘₯ ) ≀ πœ† ( π‘₯ ) . ( 3 . 2 ) It follows from (3.2) that πœ‘ 2 is unique 𝜎 -mean if and only if 𝒱 𝜎 = ξ€½ π‘₯ ∈ β„³ 𝑒 ξ€Ύ ∢ π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 β„³ 𝑒 to define absolute invariant mean in terms of a suitable sublinear functional 𝑄 on β„³ 𝑒 .

Definition 3.4. A linear functional πœ“ 2 on β„³ 𝑒 is an absolute invariant mean ( π’œ ℐ β„³ ) if and only if βˆ’ 𝑄 ( βˆ’ π‘₯ ) ≀ πœ“ 2 ( π‘₯ ) ≀ 𝑄 ( π‘₯ ) and is unique π’œ ℐ β„³ if and only if ξ€· c l ℬ 𝒱 𝜎 ξ€Έ = ξ€½ π‘₯ ∈ β„³ 𝑒 ξ€Ύ ∢ 𝑄 ( π‘₯ ) = βˆ’ 𝑄 ( βˆ’ π‘₯ ) , ( 3 . 4 ) where 𝑄 ( π‘₯ ) = l i m s u p 𝑝 , π‘ž s u p ∞ 𝑠 , 𝑑  ∞ 𝑖 = 𝑝  𝑗 = π‘ž | | πœ™ 𝑖 𝑗 𝑠 𝑑 ( | | π‘₯ ) < ∞ . ( 3 . 5 )

We have the following result.

Theorem 3.5. One has ℬ 𝒱 𝜎 = ξ€½ π‘₯ ∈ β„³ 𝑒 ξ€Ύ ∢ 𝑄 ( π‘₯ ) = 0 . ( 3 . 6 )

Proof. Since 𝑄 is a sublinear functional on β„³ 𝑒 , it follows from Hahn-Banach Theorem that there exists a continuous linear functional πœ‡ on β„³ 𝑒 such that πœ‡ ( π‘₯ ) ≀ 𝑄 ( π‘₯ ) βˆ€ π‘₯ ∈ β„³ 𝑒 , ( 3 . 7 ) and this limit is unique if and only if 𝑄 ( π‘₯ ) = βˆ’ 𝑄 ( βˆ’ π‘₯ ) = βˆ’ 𝑄 ( π‘₯ ) , that is, if and only if 𝑄 ( π‘₯ ) = 0 for all π‘₯ ∈ β„³ 𝑒 . That is, if and only if l i m ∞ 𝑝 , π‘ž  ∞ 𝑖 = 𝑝  𝑗 = π‘ž | | πœ™ 𝑖 𝑗 𝑠 𝑑 ( | | π‘₯ ) = 0 u n i f o r m l y i n 𝑠 , 𝑑 , ( 3 . 8 ) that is, if and only if π‘₯ ∈ ℬ 𝒱 𝜎 .

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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