P A M L December 2013
PURE AND APPLIED MATHEMATICS LETTERS An International Journal A characterization of weak statistical convergence via lacunary sequences and ideals
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Pure and Applied Mathematics Letters 1(2013)25-33
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A characterization of weak statistical convergence via lacunary sequences and ideals Meenakshi1, M. S. Saroa1 1
Department of Mathematics, Maharishi Markandeshwar University, Mullana Ambala, Haryana, India.
Abstract For any lacunary sequence π = (ππ ) and an ideal πΌ β π(β), we intend in present paper to define and study certain new generalizations of weak convergence which we call ππ(πΌ) βconvergence and πππ (πΌ) βconvergence on a Banach space π. Later, notions of πππ (πΌ) βlimit points and πππ (πΌ) βcluster points are also defined and some specific relations are obtained.
Keywords: weak convergence, statistical convergence, lacunary statistical convergence, ideal convergence.
AMS subject classification: 40A05; 40A35; 46B10; 46B99. Article Information: Received 07.07.2013
Revised 28.12.2013
Accepted 30.12.2013
Communicated by Prof. S. S. Bhatia
1. Introduction and Background Fast [8] and Schonenberg [23] independently introduced statistical convergence in order to assign limit to those sequences which are not convergent in the usual sense with the help of asymptotic density of subsets of β, the set of positive integers. Definition 1.1 [8] A sequence π₯ = π₯π of numbers is said to be statistically convergent to a number πΏ provided that, for every π > 0, 1 lim π β€ π: π₯π β πΏ β₯ π = 0, π π where {β
} denotes the cardinality of the enclosed set. In this case, we write π β limπββ π₯π = πΏ. In past years, many problems arising naturally in analysis have been solved by use of statistical convergence. But the quick process of change came after the works of Ε alΓ‘t [22], Fridy [9] and Connor [3]. This concept has been discussed in many fields. For instance in trigonometric series [24], summability theory [12], intuitionistic fuzzy normed spaces [16], probabilistic normed spaces [15], locally convex spaces [20] and Banach spaces [17]. In [11], Fridy and Orhan generalized statistical convergence to lacunary statistical convergence with the help of a lacunary sequence. By a lacunary sequence, we mean an increasing sequence π = (ππ ) of positive integers such that π0 = 0 and βπ = ππ β ππβ1 β β as π β β. The intervals determined by π will be denoted by πΌπ = (ππβ1 , ππ ] and the ratio ππ /ππβ1 is denoted by ππ . Definition 1.2 [11] For a lacunary sequence π = ππ , the sequence π₯ = π₯π of numbers is said to be lacunary statistically convergent to a number πΏ provided that, for every π > 0, πΏπ π β πΌπ : π₯π β πΏ β₯ π where πΏπ (πΎ) = limπ
1 βπ
= 0,
{π β πΌπ : π β πΎ} . In this case, we write ππ β limπ ββ π₯π = πΏ.
Definition 1.3 [11] A sequence π₯ = (π₯π ) of numbers is said to be ππ βconvergent to πΏ provided that, for each π > 0, lim π
1 βπ
β|π₯π β πΏ| = 0. πβπΌπ
In this case, we write ππ β limπββ π₯π = πΏ.
______________________ Corresponding author: Meenakshi, Mob. No. +91 8950651607, E-mail addresses:
[email protected] 25
Connor ππ‘ ππ. [5], have generalized statistical convergence to weak statistical convergence and given the description of Banach spaces with separable duals using weak statistical convergence. Let π be a Banach space and (π₯π ) be a sequence in π. Definition 1.4 [5] A sequence (π₯π ) is said to be weak statistically convergent to π₯ β π provided that, for each π β π β ,(π β is the continuous dual of X), each π > 0, πΏ {π β€ π: |π(π₯π β π₯)| β₯ π} = 0. In this case, we write ππ β limπββ π₯π = π₯. Although, Bhardwaj and Bala [2] have introduced weak statistical Cauchy sequences in a normed space and weak statistical convergence in ππ spaces. Bala [1], has generalized weak statistical convergence to weak* statistical convergence. In 2010, F. Nuray [21], has generalized weak statistical convergence to lacunary weak statistical convergence as follows. Definition 1.5 [21] A sequence (π₯π ) is said to be weak lacunary statistically convergent to π₯ β π provided that, for each π β π β , each π > 0, πΏπ {π β πΌπ : |π(π₯π β π₯)| β₯ π} = 0. In this case, we write πππ β limπββ π₯π = π₯. Definition 1.6 [21] A sequence
π₯π
is said to be weak ππ βconvergent to π₯ β π provided that, for each π β π β , each π > 0, 1 βπ
lim π
β π π₯π β π₯
=0
πβπΌπ
We next give a brief description of πΌ βconvergence. Kostyrko, Ε alΓ‘t and Wilczynski [18] have defined πΌ βconvergence as a natural generalization of statistical convergence. For the set of positive integers β, let π(β) denotes the power set of β. Definition 1.7 A family of sets πΌ β π(β) is called an ideal in β if and only if (i) (ii) (iii)
β
β πΌ; For each π΄, π΅ β πΌ we have π΄ βͺ π΅ β πΌ; For π΄ β πΌ and π΅ β π΄ we have π΅ β πΌ.
Definition 1.8 A non-empty family of sets πΉ β π(β) is called a filter on β if and only if (i) (ii) (iii)
β
β πΉ; For each π΄, π΅ β πΉ we have π΄ β© π΅ β πΉ; For π΄ β πΉ and π΅ β π΄ we have π΅ β πΉ.
An ideal πΌ is called non-trivial if πΌ β π(β). It immediately implies that πΌ β π(β) is a non-trivial ideal if and only if the class πΉ = πΉ(πΌ) = {β β π΄: π΄ β πΌ} is a filter on β. The filter πΉ = πΉ(πΌ) is called the filter associated with the ideal πΌ. A non-trivial ideal πΌ β π(β) is called an admissible ideal in β if and only if it contains all singletons i.e., if it contains {{π₯}: π₯ β β}. An admissible ideal πΌ β π(β) is said to be satisfy the condition (π΄π) if for every countable family of mutually disjoint sets {π΄1 , π΄2 . . . } belonging to πΌ there exists a countable family {π΅1 , π΅2 . . . } in πΌ such that π΄π π₯π΅π is a finite set for each π β β and π΅ =βͺβ π=1 π΅π β πΌ. Definition 1.9 A sequence π₯ = (π₯π ) of numbers is said to be πΌ βconvergent to a number πΏ provided that, for every π > 0, π β β: |π₯π β πΏ| β₯ π β πΌ Further, Das ππ‘ ππ. [6] extended the notion of πΌ βconvergence to πΌ βstatistical convergence and πΌ βlacunary statistical convergence as follows. Definition 1.10 A sequence π₯ = (π₯π ) of numbers is said to be πΌ βstatistically convergent to πΏ provided that, for every π > 0 and each πΎ > 0, π β β:
1 {π β€ π: |π₯π β πΏ| β₯ π} β₯ πΎ β πΌ π
Moreover, above definition is also generalized for lacunary sequences as follows. Definition 1.11 Let π = (ππ ) be a lacunary sequence as defined above. A sequence π₯ = (π₯π ) of numbers is said to be πΌ βlacunary statistically convergent to πΏ or ππ (πΌ) βconvergent to πΏ provided that, for every π > 0 and each πΎ > 0, π β β:
1 {π β πΌπ : |π₯π β πΏ| β₯ π} β₯ πΎ β πΌ. βπ
In this case we write π₯π β πΏ(ππ (πΌ)). Definition1.12 A sequence π₯ = (π₯π ) of numbers is said to be ππ (πΌ) β convergent to πΏ provided that, for every π > 0,
π β β:
1 {π β πΌπ : βπ
β|π₯π β πΏ)| β₯ π}
β πΌ.
πβπΌπ
In this case we write π₯π β πΏ(ππ (πΌ)). Some further applications of πΌ βconvergence can be found in [13] ,[14] and [19]. Throughout this paper, π will represent a Banach space, π = (ππ ) be a lacunary sequence as defined above, π₯ = (π₯π ) be a sequence of elements of π and πΌ β π(β) be a non trivial admissible ideal (as otherwise indicated). 26
2. Weak π° β lacunary statistical convergence Nuray [21], defined weak lacunary statistical convergence and proved some relations between weak lacunary statistical convergence and weak statistical convergence. In this section, we generalized these notions and study relevent connections with the help of ideals. Definition 2.1 A sequence (π₯π ) is said to be weak statistically convergent to π₯ with respect to πΌ in π if for every π > 0, every πΎ > 0 and each π β π β , where π β is the continuous dual of π we have 1 π β β: |{π β€ π: |π(π₯π β π₯)| β₯ π}| β₯ πΎ β πΌ. π In this case we write ππ(πΌ) β limπ₯π = π₯. We will denote the set of all weak πΌ β statistically convergent sequences by ππ(πΌ). For πΌ = πΌπππ , weak πΌ βstatistical convergence coincides with weak statistical convergence of [2]. Definition 2.2 A sequence (π₯π ) is said to be weak lacunary statistically convergent to π₯ with respect to πΌ in π if for every π > 0, every πΎ > 0 and each π β π β , where π β is the continuous dual of π we have π β β:
1 |{π β πΌπ : |π(π₯π β π₯)| β₯ π}| β₯ πΎ β πΌ. βπ
In this case we write πππ (πΌ) β limπ₯π = π₯. We will denote the set of all weak πΌ β lacunary statistically convergent sequences by πππ (πΌ). For πΌ = πΌπππ , weak πΌ βlacunary statistical convergence coincides with weak lacunary statistical convergence of [21]. Definition 2.3 A sequence (π₯π ) is said to be weak ππ βconvergent to π₯ with respect to πΌ in π if for every π > 0 and each π β π β , π β β:
1 βπ
β|π(π₯π β π₯)| β₯ π β πΌ. πβπΌπ
In this case we write πππ (πΌ) β limπ₯π = π₯. We will denote the set of all weak ππ (πΌ) βconvergent sequences by πππ (πΌ). For πΌ = πΌπππ , weak ππ (πΌ) βconvergence coincides with weak ππ βconvergence of [21]. Theorem 2.1 Let πΌ β π(β) be an admissible ideal. If πππ β limπ π₯π = π₯, then πππ (πΌ) β limπ π₯π = π₯. Proof Suppose πππ β limπ₯π = π₯, then for each π > 0, each π β π β , lim
πββ
1 π β πΌπ : π π₯π β π₯ βπ
β₯ π = 0.
This immediately follows that for each πΎ > 0, there exists a positive integer π such that 1 π β πΌπ : π π₯π β π₯ βπ
β₯ π < πΎ for all π β₯ π
π. π. π β β:
1 |{π β πΌπ : |π(π₯π β π₯)| β₯ π}| β₯ πΎ β {1,2,3, . . . . , π β 1} βπ
Since πΌ is admissible, therefore π β β:
1 |{π β πΌπ : |π(π₯π β π₯)| β₯ π}| β₯ πΎ β πΌ. βπ
This shows that πππ (πΌ) β limπ₯π = π₯. β The converse of above result is not true in general and can be seen from the following example. Example 2.1 Let π = π00 be the normed linear space with ||. ||π (1 < π < β), πΌ be an admissible ideal, π΄ β πΌ is fixed and π = (2π ) be a lacunary sequence as defined above. Define a sequence (π₯π ) in π00 by π, (π) π₯π
=
π, 0,
2πβ1 + 1 β€ π β€ 2πβ1 + [ βπ ]
ππ π β€ π, ππ π β€ π, ππ‘βπππ€ππ π,
2
πβ1
πππ π β π΄;
π
β€ π β€ 2 πππ π β π΄;
where πΌπ = (2πβ1 , 2π ]. For arbitrary π β π β , there is unique π¦ β ππ such that β (π)
π(π₯π ) =
βπ₯π
π¦π β€β₯ π₯ β₯π β₯ π¦ β₯π
(ππ¦ π»πππππ πππππ’ππππ‘π¦)
(1)
π =1
So, for each π > 0 (0 < π < 1) and each π β π β , 1 π β πΌπ : |π(π₯π β 0)| β₯ π βπ
β€
1 π β πΌπ : β₯ π₯ β₯π β₯ π¦ β₯π β₯ π βπ 27
=
1 βπ
π β₯ π¦ β₯π
π β πΌπ : β₯ π₯ β₯π β₯
=
1 βπ
(π)
π β πΌπ : π₯π
= π
=
[ βπ ] β 0 as π β β and π β π΄ βπ
Thus, for each πΎ > 0,
π β β:
1 π β πΌπ : |π(π₯π β 0)| β₯ π βπ
β₯ πΎ β π΄ βͺ {1,2, β¦ , π}
for some π β β. πππ (πΌ)
Since π΄ β πΌ and πΌ is admissible, it gives that π₯π
β
0. Also by (1) we can see that,
β (π)
π(π₯π ) =
βπ₯π
π¦(π) β€β₯ π₯ β₯π β₯ π¦ β₯π
(by Holder inequality)
π =1
1 π
β
=
β
β π¦(π)
π =1
1 π
β
1 π
β
(π) π π₯π
β π¦(π)
π =1
π
(by structure of the sequence)
π=1
1 π
π
β( π)π
=
π
π =1
π
=
1 π
β
(π) π π₯π
1
1 1 1 + π ππ .
ππ
(for some positive constant π) = π 2
π =1
This gives that, 1 βπ
β|π(π₯π β 0)| β€ πβπΌπ
πππ
1 βπ
1 1 1 + π ππ
βπ 2
β 0 ππ π β β,
πβπΌπ
πππ
π. π., π₯π β 0. Hence, by Theorem [4 ] of [21], we have π₯π β 0. β Theorem 2.2 For a lacunary sequence π = (ππ ), the sequence (π₯π ) is weak ππ (πΌ) βconvergent to π₯ if and only if (π₯π ) is weak πΌ βlacunary statistically convergent to π₯. Proof Let (π₯π ) is weak ππ (πΌ) βconvergent to π₯, then for every π > 0 and each π β π β , we have π β β:
1 βπ
β|π(π₯π β π₯)| β₯ π β πΌ. πβπΌπ
Also for given π > 0, we have 1 βπ
β|π(π₯π β π₯)| β₯ πβπΌπ
β₯
1 βπ
β π(π₯π β π₯) πβπΌπ π(π₯ π βπ₯) β₯π
π {π β πΌπ : π(π₯π β π₯) β₯ π} . βπ
Hence, for every π > 0, each πΎ > 0 and each π β π β we have π β β:
1 1 |{π β πΌπ : |π(π₯π β π₯)| β₯ π}| β₯ πΎ β π β β: βπ βπ
β|π(π₯π β π₯)| β₯ ππΎ β πΌ. πβπΌπ
This immediately gives (π₯π ) is weak πΌ βlacunary statistically convergent to π₯. Conversely, suppose that (π₯π ) is weak πΌ βlacunary statistically convergent to π₯. Then for every π > 0, every πΎ > 0 and each π β π β we have
π β β:
1 |{π β πΌπ : |π(π₯π β π₯)| β₯ π}| β₯ πΎ β πΌ. βπ
Since π β π β , therefore π is bounded. Let |π(π₯π β π₯)| β€ π for all π. For given π > 0, we have 28
1 βπ
β|π(π₯π β π₯)| = πβπΌπ
β€
1 βπ
β π(π₯π β π₯) + πβπΌπ π(π₯ π βπ₯) β₯π
β π(π₯π β π₯) πβπΌπ π(π₯ π βπ₯) 1. Proof Let liminfπ ππ > 1, then there exists a πΏ > 0 such that ππ > 1 + πΏ for sufficiently large π which implies that β
ππ βπ
β€
πΏ+1 πΏ
. Let ππ(πΌ) β
limπ₯π = π₯, then for every π > 0, every πΎ > 0 and each π β π , we have 1 π β β: |{π β€ π: |π(π₯π β π₯)| β₯ π}| β₯ πΎ β πΌ. π Also, one can easily observe that 1 1 {π β€ ππ : |π(π₯π β π₯)| β₯ π} β₯ {π β πΌπ : |π(π₯π β π₯)| β₯ π} ππ ππ βπ 1 {π β πΌπ : |π(π₯π β π₯)| β₯ π} ππ βπ
= β₯
πΏ
1
πΏ+1 β π
{π β πΌπ : |π(π₯π β π₯)| β₯ π} ;
So, π β β:
1 π β πΌπ : π π₯π β π₯ βπ
β₯π
β₯ πΎ β π β β:
1 πΎπΏ {π β€ ππ : |π(π₯π β π₯)| β₯ π} β₯ β πΌ, ππ πΏ+1
which gives the result. Conversely, suppose that liminfπ ππ = 1, then following as in [11], one can find a subsequence (ππ(π ) ) of π = (ππ ) such that ππ(π ) 1 < 1+ ππ(π )β1 π
πππ
ππ(π )β1 >π ππ(π β1)
π€βπππ
π(π) β₯ π(π β 1) + 2
Define a sequence (π₯π ) as follows
π₯π =
1, 0,
ππ π β πΌπ (π ) , ππ‘βπππ€ππ π .
Since π β π β , therefore π is bounded. Let |π(π₯π β π₯)| β€ π for all π. Thus for any π β π β , π₯ β π, 1 βπ(π )
β|π(π₯π β π₯)| = πβπΌπ(π )
1 βπ(π )
β|π(1 β π₯)| β€ π πππ π = 1,2,β
β
β
πβπΌπ(π )
and 1 βπ
β|π(π₯π β π₯)| = πβπΌπ
1 βπ
β|π(βπ₯)| β€ π πππ π β ππ πβπΌπ
which gives (π₯π ) β πππ (πΌ). By Theorem 2.2 we have (π₯π ) β πππ (πΌ). Let ππ(π β1) β€ π β€ ππ(π ) , then 1 π
π
π=1
1 β|π(π₯π β π₯)| β₯ π
β₯
π
β π(π₯π β π₯) π=1 π(π₯ π βπ₯) β₯π
π {π β€ π: π(π₯π β π₯) β₯ π} . π
We can write, 29
π 1 {π β€ π: π(π₯π β π₯) β₯ π} β€ π π
π
π=1
1 1 2π β|π(π₯π β π₯)| β€ π( + ) = π π π
Thus, (π₯π ) is weak πΌ β statistically convergent to π₯. β In [21] it has been shown that for any lacunary sequence π = (ππ ), πππ β ππ if and only if limsupπ ππ < β. However, for any admissible ideal πΌ this is not sure in general. Theorem 2.4 Let πΌ be an admissible ideal having condition (AP) and π β πΉ(πΌ). If π₯ β ππ(πΌ) β© πππ (πΌ), then ππ(πΌ) β limπ₯π = πππ (πΌ) β limπ₯π . Proof Suppose ππ(πΌ) β limπ₯π = π₯ and πππ (πΌ) β limπ₯π = π¦ and π₯ β π¦. Let |π₯ β π¦| > 2π, (π > 0). Since πΌ satisfy the condition (AP) so there exists π β πΉ(πΌ) such that lim
πββ
1 {π β€ ππ : π(π₯π β π₯) β₯ π} = 0, ππ
where π = π1 , π2 , π3 . . . . . Let, πΊ = π β€ ππ : π π₯π β π₯
β₯ π and
π» = π β€ ππ : π π₯π β π¦ Since limπββ
|πΊ| ππ
= 0, it follows that limπββ
1 ππ
β₯π .
{π β€ ππ : π(π₯π β π¦) β₯ π} = 1.
π‘β Let, π β = ππ1 , ππ2 , ππ3 , β¦ = π β© π β πΉ(πΌ). Then the πππ term of the weak statistical limit expression
1 1 {π β€ ππ : π(π₯π β π¦) β₯ π} : ππ ππ π
ππ
πβ
1 = ππ π
βπΌπ : π(π₯π β π¦) β₯ π π=1
= where π‘π =
1 βπ
ππ
β {π β πΌπ : π(π₯π β π¦) β₯ π} π =1
1
ππ
ββπ π‘π ππ β β π π=1 π=1
(2)
{π β πΌπ : π(π₯π β π¦) β₯ π} and π‘π βΆπΌ 0 as πππ (πΌ) β limπ₯π = π¦. Since π is a lacunary sequence and the right side of above
expression is a regular weighted mean transform of the sequence π‘ = π‘π , therefore it is also πΌ-convergent to zero as π β β. Also πΌ satisfies 1 the condition (AP) which gives it has a subsequence converging to zero. Since this is a subsequence of {π β€ ππ : π(π₯π β π¦) β₯ π} , we infer that
1 ππ
ππ
{π β€ ππ : π(π₯π β π¦) β₯ π} is not convergent to 1 which is a contradiction. This completes the proof of the theorem. β
3. Weak π° βlacunary statistical limit and cluster point Fridy [10] introduced the notion of statistical limit point and statistical cluster point for the sequences of numbers. Connor ππ‘ ππ. [4] generalised these notions to π΄ βstatistical limit point and π΄ βstatistical cluster point where π΄ is a nonnegative regular summability matrix. Demirci [7] generalised these notions to π βstatistical limit point and π βstatistical cluster point for sequences of numbers. GΓΌrdal ππ‘ ππ. [14] have generalized the notion of π΄ βstatistical limit point and π΄ βstatistical cluster point to define π΄πΌ βstatistical limit point and π΄πΌ βstatistical cluster point where πΌ β π(β) is an ideal. Further these concepts has been defined in different spaces. Now we will introduce the notion of weak πΌ βlacunary statistical limit and cluster point in Banach spaces. Let π = ππ be a lacunary sequence. For a Banach space π, let π₯ = π₯π be a sequence in π. Let π₯π πΎ = π π : π β β , then we denote π₯π π by (π₯)πΎ . If for every πΎ > 0, π β β:
π
be a subsequence of π₯ and
1 π π β πΌπ : π β β β₯ πΎ β πΌ; βπ
then (π₯)πΎ is called π πΌ -thin subsequence. on the other hand (π₯)πΎ is a π πΌ -nonthin subsequence of π₯ provided that π β β:
1 π π β πΌπ : π β β β₯ πΎ β πΌ. βπ
Definition 3.1 Let π be a Banach space, π = (ππ ) be a lacunary sequence and π β π β . An element π₯ β π is called a weak ππ (πΌ) βlimit point of a sequence π₯ = (π₯π ) in π provided that there is a π(πΌ)-nonthin subsequence of (π₯π ) that is weak convergent to π₯. Let Ξ(πππ (πΌ), π₯) denotes the set of all πππ (πΌ)-limit points of the sequence π₯ = (π₯π ). Definition 3.2 Let π be a Banach space, π = ππ be a lacunary sequence and π β π β . An element π₯ β π is called a weak ππ πΌ βcluster point of a sequence π₯ = π₯π in π provided that for every π > 0 and each π β π β , we have π β β:
1 π β πΌπ : π π₯π β π₯ βπ
0, π β β:
1 {π π β πΌπ : π β β} β₯ πΎ β πΌ βπ
(3)
Now, the containment π β πΌπ : π(π₯π β π₯) < π β {π(π) β πΌπ : π(π₯π(π ) β π₯) < π} gives π β πΌπ : π(π₯π β π₯) < π β π π β πΌπ : π β β β π π β πΌπ : π(π₯π (π ) β π₯) β₯ π ; which immediately implies 1 π β πΌπ : π(π₯π β π₯) < π βπ
β₯
1 1 π π β πΌπ : π β β β π π β πΌπ : π(π₯π(π ) β π₯) β₯ π . βπ βπ
So, for each πΎ > 0 and π > 0, π β β:
1 π β πΌπ : π(π₯π β π₯) < π βπ
β₯ πΎ β π β β: π π β πΌπ : π β β β₯ πΎ β π β β: π π β πΌπ : π(π₯π (π ) β π₯) β₯ π
= π β β: π π β πΌπ : π β β β₯ πΎ β© π β β: π π β πΌπ : π(π₯π(π ) β π₯) β₯ π
β₯πΎ
= π β β: π π β πΌπ : π β β β₯ πΎ β© π β β: π π β πΌπ : π(π₯π (π ) β π₯) β₯ π
0 and each π β π β , the set π π β πΌπ : π(π₯π (π ) β π₯) β₯ π is finite for which we have limπββ
1 π π β πΌπ : π(π₯π(π ) β π₯) β₯ π βπ
=0,
π. π. πππ β limπ₯π (π ) = π₯. Using Theorem 2.1, we get πππ (πΌ) β limπ₯π(π ) = π₯. Thus, for each π > 0, each πΎ > 0 and each π β π β π β β:
1 |π β πΌπ : |π(π₯π β π₯)| β₯ π| β₯ πΎ β πΌ βπ
or
π β β:
1 |π β πΌπ : |π(π₯π β π₯)| β₯ π| < πΎ β πΉ πΌ βπ
(5)
Using (3) and (5) in (4), we get π β β:
1 βπ
π β πΌπ : π(π₯π β π₯) < π
β₯ πΎ β πΉ(πΌ).
π. π. π β β:
1 π β πΌπ : π(π₯π β π₯) < π βπ
β₯ πΎ β πΌ.
This shows that π₯ β Ξ(πππ (πΌ), π₯) and therefore we have the containment Ξ(πππ (πΌ), π₯) β Ξ(πππ (πΌ), π₯). β Theorem 3.2 Let π = ππ be a lacunary sequence. For any sequence π₯ = (π₯π ) in a Banach space π, if πππ (πΌ) β limπ π₯π = π₯0 , then Ξ πππ (πΌ), π₯ = Ξ πππ (πΌ), π₯ = {π₯0 }. Proof. We first show that Ξ πππ (πΌ) , π₯ = π₯0 . Let Ξ πππ (πΌ), π₯ = π₯0 , π¦0 such that π₯0 β π¦0 . By definition there exist two π(πΌ)-nonthin subsequences π₯π(π) and π₯π(π ) of the sequence π₯ = (π₯π ) which are respectively weakly convergent to π₯0 and π¦0 . Choose π > 0 such that |π(π₯ )βπ(π¦ )|
0 0 0 0 and πΎ > 0, the set π β β: = π β β:
1 π π β πΌπ : π(π₯π(π ) β π¦0 ) < π βπ
β₯πΎ
1 1 π π β πΌπ : π β β β₯ πΎ β π β β: π π β πΌπ : π(π₯π(π ) β π¦0 ) β₯ π βπ βπ
β₯πΎ 31
= π β β:
1 1 π π β πΌπ : π β β β₯ πΎ β© π β β: π π β πΌπ : π(π₯π(π ) β π¦0 ) β₯ π βπ βπ
π
β₯πΎ .
Since π(π) is π(πΌ) βnonthin subsequence and πππ (πΌ) β limπ₯π(π ) = π¦0 , therefore π β β:
1 1 π π β πΌπ : π β β β₯ πΎ β© π β β: π π β πΌπ : π(π₯π(π ) β π¦0 ) β₯ π βπ βπ
π
β₯πΎ
β πΉ(πΌ).
This gives, π β β:
1 π π β πΌπ : π(π₯π(π ) β π¦0 ) < π βπ
β₯πΎ βπΉ πΌ
(6)
Also πππ (πΌ) β limπ π₯π = π₯0 , so we have π β β:
1 π π β πΌπ : π(π₯π β π₯0 ) β₯ π βπ
β₯πΎ βπΌ
(7)
For 0 < 2π < π π₯0 β π¦0 , π π β πΌπ : π(π₯π(π ) β π¦0 ) < π β© π β πΌπ : π(π₯π β π₯0 ) < π = β
so we have, π π β πΌπ : π(π₯π(π ) β π¦0 ) < π β {π β πΌπ : π(π₯π β π₯0 ) β₯ π}; which implies immadetaly that 1 π π β πΌπ : π(π₯π(π ) β π¦0 ) < π βπ
β€
1 {π β πΌπ : π(π₯π β π₯0 ) β₯ π} . βπ
For π > 0, πΎ > 0 π β β:
1 π π β πΌπ : π(π₯π(π ) β π¦0 ) < π βπ
β₯ πΎ β π β β:
1 π β πΌπ : π(π₯π β π₯0 ) β₯ π} β₯ πΎ βπ
Using(7), we have π β β:
1 π π β πΌπ : π(π₯π(π ) β π¦0 ) < π βπ
β₯ πΎ β πΌ;
which contradict(6). Hence, Ξ πππ (πΌ), π₯ = π₯0 . Similarly, we can show that Ξ πππ (πΌ), π₯ = π₯0 . β Theorem 3.3 If π₯ = π₯π
and π¦ = (π¦π )
are two sequences in π such that limπ
1 βπ
π β πΌπ : π₯π β π¦π
= 0, then Ξ πππ (πΌ), π₯ =
Ξ πππ (πΌ), π¦ and Ξ πππ (πΌ), π₯ = Ξ πππ (πΌ), π¦ . Proof Suppose π₯ β² β Ξ πππ (πΌ), π₯ , then there exists a π(πΌ) βnonthin subsequence (π₯)πΎ of the sequence π₯ = (π₯π ) that weakly converges to π₯ β² . π. π. If πΎ = π π : π β β , then for every πΎ > 0, π β β: Since, limπ
1 βπ
1 {π π β πΌπ : π β β} β₯ πΎ β πΌ πππ limπ(π₯π(π ) β π₯ β² ) = 0 π βπ
{π β πΌπ : π β πΎ, π₯π β π¦π } = 0, it follows that limsup π
1 {π β πΌπ : π β πΎ, π₯π = π¦π } > 0 βπ
(8)
Therefore, there exists a π(πΌ) βnonthin subsequence (π¦)πΎ of the sequence π¦ = (π¦π ) that weakly converges to π₯ β² . This shows that π₯ β² β Ξ πππ (πΌ), π¦ and therefore Ξ πππ (πΌ), π₯ β Ξ πππ (πΌ), π¦ . By symmetry we have Ξ πππ (πΌ), π¦ β Ξ πππ (πΌ), π₯ . Hence, Ξ πππ (πΌ), π₯ = Ξ πππ (πΌ), π¦ . Similarly we can prove Ξ πππ (πΌ), π₯ = Ξ πππ (πΌ), π¦ . β
Acknowledgement The authors are thankful to the reviewers for their valuable comments and suggestions.
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