Page 1

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 6 (May. - Jun. 2013), PP 01-04 www.iosrjournals.org

Matrix Transformations on Some Difference Sequence Spaces Z. U. Siddiqui, A. Kiltho Department of Mathematics and Statistics, University of Maiduguri, Nigeria

Abstract: The sequence spaces đ?‘™âˆž (đ?‘˘, đ?‘Ł, ∆), đ?‘?0 (đ?‘˘, đ?‘Ł, ∆) and đ?‘?(đ?‘˘, đ?‘Ł, ∆) were recently introduced. The matrix

classes (đ?‘? đ?‘˘, đ?‘Ł, ∆ : đ?‘?) and (đ?‘? đ?‘˘, đ?‘Ł, ∆ : đ?‘™âˆž ) were characterized. The object of this paper is to further determine the necessary and sufficient conditions on an infinite matrix to characterize the matrix classes (đ?‘? đ?‘˘, đ?‘Ł, ∆ âˆś đ?‘?đ?‘ ) and (đ?‘? đ?‘˘, đ?‘Ł, ∆ âˆś đ?‘™đ?‘? ). It is observed that the later characterizations are additions to the existing ones. Keywords- Difference operators, Duals, Generalized weighted mean, Matrix transformations

I.

Introduction

The sequence spaces đ?&#x2018;&#x2122;â&#x2C6;&#x17E; (â&#x2C6;&#x2020;), đ?&#x2018;?0 (â&#x2C6;&#x2020;) and đ?&#x2018;?(â&#x2C6;&#x2020;) were first introduced by Kizmaz [6] in 1981. Similar to the sequence spaces đ?&#x2018;&#x2122;â&#x2C6;&#x17E; (đ?&#x2018;?), đ?&#x2018;?0 (đ?&#x2018;?) and đ?&#x2018;?(đ?&#x2018;?) for đ?&#x2018;?đ?&#x2018;&#x2DC; > 1 of Maddox [7] and Simons [10], the â&#x2C6;&#x2020;- sequence spaces above were extended to â&#x2C6;&#x2020;đ?&#x2018;&#x2122;â&#x2C6;&#x17E; (đ?&#x2018;?), â&#x2C6;&#x2020;đ?&#x2018;?0 (đ?&#x2018;?) and â&#x2C6;&#x2020;đ?&#x2018;?(đ?&#x2018;?) by Ahmad and Mursaleen [1] in â&#x20AC;Ś The concept of difference operators has been discussed and used by Polat and BaĹ&#x;ar [8] and by Altay and BaĹ&#x;ar [2], both in 2007. The idea of generalized weighted mean was applied by Altay and BaĹ&#x;ar [3], in 2006. This concept depends on the idea of đ??ş đ?&#x2018;˘, đ?&#x2018;Ł - transforms which has been used by Polat, et al [10] and by Basarir and Kara [4]. We shall need the following sequence spaces: đ?&#x153;&#x201D; = {đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;ś x is any sequence } đ?&#x2018;? = {đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D;: đ?&#x2018;Ľđ?&#x2018;&#x2DC; converges, i.e. limđ?&#x2018;&#x2DC;â&#x2020;&#x2019;â&#x2C6;&#x17E; đ?&#x2018;Ľđ?&#x2018;&#x2DC; exists } đ?&#x2018;?0 = đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D;: limkâ&#x2020;&#x2019;â&#x2C6;&#x17E; đ?&#x2018;Ľđ?&#x2018;&#x2DC; = 0 , the set of all null sequences đ?&#x2018;&#x2122;â&#x2C6;&#x17E; = đ?&#x2018;&#x161; = đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D;: đ?&#x2018;Ľ â&#x2C6;&#x17E; = đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x203A; đ?&#x2018;Ľđ?&#x2018;&#x2DC; < â&#x2C6;&#x17E; đ?&#x2018;&#x2122;1 = đ?&#x2018;&#x2122; = đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D;: đ?&#x2018;Ľ 1 = â&#x2C6;&#x17E;đ?&#x2018;&#x2DC;=0 đ?&#x2018;Ľđ?&#x2018;&#x2DC; < â&#x2C6;&#x17E; đ?&#x2018;&#x2122;đ?&#x2018;? = đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D;: đ?&#x2018;Ľ đ?&#x2018;? = đ?&#x2018;Ľđ?&#x2018;&#x2DC; đ?&#x2018;? < â&#x2C6;&#x17E;; 1 â&#x2030;¤ đ?&#x2018;? < â&#x2C6;&#x17E; đ?&#x153;&#x2122; = {đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D;: â&#x2C6;&#x192; đ?&#x2018; â&#x2C6;&#x2C6; â&#x201E;&#x2022;đ?&#x2018;&#x2022;such that â&#x2C6;&#x20AC; đ?&#x2018;&#x2DC; â&#x2030;Ľ đ?&#x2018; , đ?&#x2018;Ľđ?&#x2018;&#x2DC; = 0}, the set of finitely nonzero sequences đ?&#x2018;?đ?&#x2018;  = đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D;: đ?&#x2018;Ľ đ?&#x2018;?đ?&#x2018;  = đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC;=0 đ?&#x2018;Ľđ?&#x2018;&#x2DC; < â&#x2C6;&#x17E; , the set of all sequences with bounded partial sums đ?&#x2018;&#x2039;đ?&#x203A;˝ = {đ?&#x2018;&#x17D; = đ?&#x2018;&#x17D;đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D;: â&#x2C6;&#x17E;đ?&#x2018;&#x2DC;=0 đ?&#x2018;&#x17D;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x2018;? , â&#x2C6;&#x20AC; đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;} Note that đ?&#x2018;Ľ = (đ?&#x2018;Ľđ?&#x2018;&#x2DC; ) is used throughout for the convention (đ?&#x2018;Ľđ?&#x2018;&#x2DC; ) = (đ?&#x2018;Ľđ?&#x2018;&#x2DC; )â&#x2C6;&#x17E;đ?&#x2018;&#x2DC;=0 . We take đ?&#x2018;&#x2019; = (1, 1, 1, â&#x20AC;Ś ) and đ?&#x2018;&#x2019; đ?&#x2018;&#x2DC; for the sequence whose only nonzero term is 1 in the đ?&#x2018;&#x2DC;th place for each đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; â&#x201E;&#x2022;, where â&#x201E;&#x2022; = {0, 1, 2, 3, â&#x20AC;Ś }. Any vector subspace of đ?&#x153;&#x201D; is called a sequence subspace. A sequence space đ?&#x2018;&#x2039; is FK if it is a complete linear metric space with continuous coordinates đ?&#x2018;&#x192;đ?&#x2018;&#x203A; â&#x2C6;ś đ?&#x2018;&#x2039; â&#x2020;&#x2019; â&#x201E;&#x201A;, defined by đ?&#x2018;&#x192;đ?&#x2018;&#x203A; đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;&#x203A; â&#x2C6;&#x20AC; đ?&#x2018;Ľ = (đ?&#x2018;Ľđ?&#x2018;&#x2DC; ) â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; with đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;. A normed FK space is BK-space or Banach space with continuous coordinates. An FK space has AK- property if đ?&#x2018;Ľ [đ?&#x2018;&#x161;] â&#x2020;&#x2019; đ?&#x2018;Ľ in đ?&#x2018;&#x2039;, where đ?&#x2018;Ľ [đ?&#x2018;&#x161; ] = đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC;=0 đ?&#x2018;Ľđ?&#x2018;&#x2DC; đ?&#x2018;&#x2019; đ?&#x2018;&#x2DC; is the mth- section of đ?&#x2018;Ľ. If đ?&#x153;&#x2018; is dense in đ?&#x2018;&#x2039; then it has an AD- property (see Boos [5]). A matrix domain of a sequence space đ?&#x2018;&#x2039;, is defined as đ?&#x2018;&#x2039;đ??´ = đ?&#x2018;Ľ = (đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D; â&#x2C6;ś đ??´đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; }. 1

Let đ?&#x2019;° be the set of all sequences đ?&#x2018;˘ = (đ?&#x2018;˘đ?&#x2018;&#x2DC; ) with đ?&#x2018;˘đ?&#x2018;&#x2DC; â&#x2030; 0 â&#x2C6;&#x20AC; đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; â&#x201E;&#x2022;, and for đ?&#x2018;˘ â&#x2C6;&#x2C6; đ?&#x2019;° let đ?&#x2018;˘ =

for đ?&#x2018;˘, đ?&#x2018;Ł â&#x2C6;&#x2C6; đ?&#x2019;° define the matrix đ??ş đ?&#x2018;˘, đ?&#x2018;Ł = (đ?&#x2018;&#x201D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; ) by đ?&#x2018;˘ đ?&#x2018;Ł , for 0 â&#x2030;¤ đ?&#x2018;&#x2DC; â&#x2030;¤ đ?&#x2018;&#x203A;, đ?&#x2018;&#x201D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; = đ?&#x2018;&#x203A; đ?&#x2018;&#x2DC; 0, for đ?&#x2018;&#x2DC; > đ?&#x2018;&#x203A; â&#x2C6;&#x20AC; đ?&#x2018;&#x2DC;, đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022; This matrix is called the generalized weighted mean. The sequence đ?&#x2018;Ś = (đ?&#x2018;Śđ?&#x2018;&#x2DC; ) in the sequence spaces đ?&#x153;&#x2020; đ?&#x2018;˘, đ?&#x2018;Ł, Î&#x201D; = {đ?&#x2018;Ľ = (đ?&#x2018;Ľđ?&#x2018;&#x2DC; ) â&#x2C6;&#x2C6; đ?&#x153;&#x201D; â&#x2C6;ś đ?&#x2018;Ś = đ?&#x2018;&#x2DC;đ?&#x2018;&#x2013;=0 đ?&#x2018;˘đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2020;đ?&#x2018;Ľđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;}, đ?&#x153;&#x2020; â&#x2C6;&#x2C6; {đ?&#x2018;&#x2122;â&#x2C6;&#x17E; , đ?&#x2018;?, đ?&#x2018;?0 }

1

đ?&#x2018;˘đ?&#x2018;&#x2DC;

. Then

(1)

is the đ??ş đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; â&#x2C6;&#x2019;transform of a given sequence đ?&#x2018;Ľ = (đ?&#x2018;Ľđ?&#x2018;&#x2DC; ). It is defined by đ?&#x2018;Ś = đ?&#x2018;&#x2DC;đ?&#x2018;&#x2013;=0 đ?&#x2018;˘đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2020;đ?&#x2018;Ľđ?&#x2018;&#x2013; = đ?&#x2018;&#x2DC;đ?&#x2018;&#x2013;=0 đ?&#x2018;˘đ?&#x2018;&#x2DC; â&#x2C6;&#x2021;đ?&#x2018;Łđ?&#x2018;&#x2013; đ?&#x2018;Ľđ?&#x2018;&#x2013; where, â&#x2C6;&#x2021;đ?&#x2018;Łđ?&#x2018;&#x2013; = đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2013;+1 and â&#x2C6;&#x2020;đ?&#x2018;Ľ = â&#x2C6;&#x2020;đ?&#x2018;Ľđ?&#x2018;&#x2013; = đ?&#x2018;Ľđ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 , and taking all negative subscripts to be naught. The spaces (1) were defined in [9]. If đ?&#x2018;&#x2039; is any normed sequence space the matrix domain đ?&#x2018;&#x2039;đ??ş(đ?&#x2018;˘ ,đ?&#x2018;Ł,â&#x2C6;&#x2020;) is the generalized weighted mean difference sequence space [9]. Our object is to characterize the matrix classes đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; â&#x2C6;ś đ?&#x2018;&#x2122;đ?&#x2018;? and (đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; â&#x2C6;ś đ?&#x2018;?đ?&#x2018; ). However, matrix class characterizations are done with help of đ?&#x203A;˝ â&#x2C6;&#x2019;duals, and so we need the following www.iosrjournals.org

1 | Page


Matrix Transformations on Some Difference Sequence Spaces Lemma 1.1 [9]: Let đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2C6; đ?&#x2019;°, đ?&#x2018;&#x17D; = đ?&#x2018;&#x17D;đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D; and the matrix đ??ˇ = (đ?&#x2018;&#x2018;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; ) by 1 đ?&#x2018;˘ đ?&#x2018;&#x203A; đ?&#x2018;Łđ?&#x2018;&#x2DC; 1

đ?&#x2018;&#x2018;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; =

đ?&#x2018;˘ đ?&#x2018;&#x203A; đ?&#x2018;Łđ?&#x2018;&#x203A;

â&#x2C6;&#x2019;đ?&#x2018;˘

1 đ?&#x2018;&#x203A; đ?&#x2018;Łđ?&#x2018;&#x2DC;+1

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; ;

đ?&#x2018;&#x17D;đ?&#x2018;&#x2DC; ;

0â&#x2030;¤đ?&#x2018;&#x2DC;<đ?&#x2018;&#x203A; ,

đ?&#x2018;&#x2DC; =đ?&#x2018;&#x203A;

0; đ?&#x2018;&#x2DC;>đ?&#x2018;&#x203A; and let đ?&#x2018;&#x2018;1 , đ?&#x2018;&#x2018;2 , đ?&#x2018;&#x2018;3 , đ?&#x2018;&#x2018;4 and đ?&#x2018;&#x2018;5 be the sets đ?&#x2018;&#x2018;1 = {đ?&#x2018;&#x17D; = đ?&#x2018;&#x17D;đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D; â&#x2C6;ś đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; đ?&#x2018;&#x2DC;â&#x2C6;&#x2C6;đ?&#x2019;Ś đ?&#x2018;&#x2018;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; < â&#x2C6;&#x17E;}; đ?&#x2018;&#x2018;2 = {đ?&#x2018;&#x17D; = đ?&#x2018;&#x17D;đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D; â&#x2C6;ś đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; đ?&#x2018;&#x2018;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; < â&#x2C6;&#x17E;}; đ?&#x2018;&#x2018;3 = {đ?&#x2018;&#x17D; = đ?&#x2018;&#x17D;đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; đ?&#x153;&#x201D; â&#x2C6;ś đ?&#x2018;&#x2122;đ?&#x2018;&#x2013;đ?&#x2018;&#x161;đ?&#x2018;&#x203A; â&#x2020;&#x2019;â&#x2C6;&#x17E; đ?&#x2018;&#x2018;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; exists for each đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;} Then, [đ?&#x2018;?0 đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; ]đ?&#x203A;˝ = đ?&#x2018;&#x2018;1 â&#x2C6;Š đ?&#x2018;&#x2018;2 â&#x2C6;Š đ?&#x2018;&#x2018;3 .

II.

Methodology

If A is an infinite matrix with complex entries đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; (đ?&#x2018;&#x203A;, đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; â&#x201E;&#x2022;), then đ??´ = (đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; ) is used for đ??´ = (đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; )â&#x2C6;&#x17E;đ?&#x2018;&#x203A;,đ?&#x2018;&#x2DC; =0 and đ??´đ?&#x2018;&#x203A; is the sequence in the nth row of A, or đ??´đ?&#x2018;&#x203A; = (đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; )â&#x2C6;&#x17E;đ?&#x2018;&#x2DC;=0 for every đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;. The A- transform of a sequence x is defined as đ??´đ?&#x2018;Ľ = (đ??´đ?&#x2018;&#x203A; (đ?&#x2018;Ľ))â&#x2C6;&#x17E;đ?&#x2018;&#x203A; =0 = limđ?&#x2018;&#x203A;â&#x2020;&#x2019;â&#x2C6;&#x17E; â&#x2C6;&#x17E;đ?&#x2018;&#x2DC;=0 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;&#x2DC; (đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;) provided the series on the right converges for each n and for all đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;. The pair (đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) is referred to as a matrix class, so that đ??´đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;đ?&#x203A;˝ â&#x2C6;&#x20AC; đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022; (2) and đ??´đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x152; â&#x2C6;&#x20AC; đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;, in the norm of đ?&#x2018;&#x152; In this paper we shall take đ?&#x2018;&#x2039; = đ?&#x2018;?(đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020;) and đ?&#x2018;&#x152; â&#x2C6;&#x2C6; đ?&#x2018;&#x2122;đ?&#x2018;? , đ?&#x2018;?đ?&#x2018; . We shall need the following lemma for the proof of Theorems 3.1 and 3.2 as our main results in section 3: đ??´ â&#x2C6;&#x2C6; (đ?&#x2018;&#x2039;, đ?&#x2018;&#x152;) â&#x;ş

Lemma 2.1 [9]: The sequence spaces đ?&#x153;&#x2020; đ?&#x2018;˘, đ?&#x2018;Ł, Î&#x201D; for đ?&#x153;&#x2020; â&#x2C6;&#x2C6; {đ?&#x2018;&#x2122;â&#x2C6;&#x17E; , đ?&#x2018;?, đ?&#x2018;?0 } are complete normed linear spaces with the norm đ?&#x2018;Ľ đ?&#x153;&#x2020; đ?&#x2018;˘ ,đ?&#x2018;Ł,Î&#x201D; = supđ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2013;=0 đ?&#x2018;˘đ?&#x2018;&#x2DC; â&#x2C6;&#x2020;đ?&#x2018;Ľđ?&#x2018;&#x2013; = đ?&#x2018;Ś đ?&#x153;&#x2020; . They are also BK spaces with both AK- and AD- properties. Further, let đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;?0 and define đ?&#x2018;Ľ = đ?&#x2018;Ľđ?&#x2018;&#x2DC; by 1 1 1 1 đ?&#x2018;Ľđ?&#x2018;&#x2DC; = đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;đ?&#x2018;Ł đ?&#x2018;Śđ?&#x2018;&#x2013; + đ?&#x2018;˘ đ?&#x2018;Ł đ?&#x2018;Śđ?&#x2018;&#x2DC; ; đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; â&#x201E;&#x2022; đ?&#x2018;&#x2013;=0 đ?&#x2018;˘ đ?&#x2018;Ł đ?&#x2018;&#x2DC;

then đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;?0 đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; .

đ?&#x2018;&#x2013;

đ?&#x2018;&#x2013;+1

đ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC;

An infinite matrix A maps a BK space đ?&#x2018;&#x2039; continuously into the space đ?&#x2018;?đ?&#x2018; if and only if the sequence the sequence of functional {đ?&#x2018;&#x201C;đ?&#x2018;&#x203A; } defined by â&#x2C6;&#x17E; đ?&#x2018;&#x201C;đ?&#x2018;&#x203A; đ?&#x2018;Ľ = đ?&#x2018;&#x161; đ?&#x2018;&#x203A;=1 đ?&#x2018;&#x2DC;=1 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;&#x2DC; , đ?&#x2018;&#x203A; = 1, 2, 3, â&#x20AC;Ś is bounded in the dual space of đ?&#x2018;&#x2039;.

III. Theorem 3.1. đ??´ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; â&#x2C6;ś đ?&#x2018;&#x2122;đ?&#x2018;? (i) (ii) (iii)

đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x203A; đ?&#x2018;&#x2122;đ?&#x2018;&#x2013;đ?&#x2018;&#x161;đ?&#x2018;&#x203A; â&#x2020;&#x2019;â&#x2C6;&#x17E; đ?&#x2018;&#x2122;đ?&#x2018;&#x2013;đ?&#x2018;&#x161;đ?&#x2018;&#x203A; â&#x2020;&#x2019;â&#x2C6;&#x17E;

đ?&#x2018;&#x2DC;â&#x2C6;&#x2C6;đ?&#x2019;Ś

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2013;=1 đ?&#x2018;˘

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2013;=1 đ?&#x2018;˘ đ?&#x2018;&#x2DC; đ?&#x2018;&#x203A; đ?&#x2018;&#x2DC;=0

1 đ?&#x2018;Łđ?&#x2018;&#x2013;

đ?&#x2018;&#x2DC;

1 đ?&#x2018;Łđ?&#x2018;&#x2013; 1

â&#x2C6;&#x2019;đ?&#x2018;Ł

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2013;=1 đ?&#x2018;˘ đ?&#x2018;&#x2DC;

Main Results

for đ?&#x2018;? > 1, if and only if â&#x2C6;&#x2019;

đ?&#x2018;&#x2013;+1

1

1

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; + đ?&#x2018;˘

â&#x2C6;&#x2019;đ?&#x2018;Ł

đ?&#x2018;Łđ?&#x2018;&#x2013;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; +

đ?&#x2018;Łđ?&#x2018;&#x2013;+1

1

1 đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2DC;

1 đ?&#x2018;˘ đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2DC;

đ?&#x2018;?

< â&#x2C6;&#x17E;,

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; = đ?&#x2018;&#x17D;đ?&#x2018;&#x2DC; , exists

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; + đ?&#x2018;˘

đ?&#x2018;&#x2013;+1

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC;

1 đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2DC;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; = đ?&#x2018;&#x17D;, exists

Proof: Since đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; and đ?&#x2018;&#x2122;đ?&#x2018;? are BK spaces, we suppose that (i), (ii) and (iii) hold and take đ?&#x2018;Ľ = (đ?&#x2018;Ľđ?&#x2018;&#x2DC; ) â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; . Then by (2) and Lemma 1.1, đ??´đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; [đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; ]đ?&#x203A;˝ for all đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;, which implies the existence of the Atransform of đ?&#x2018;Ľ, or đ??´đ?&#x2018;Ľ exists for each đ?&#x2018;&#x203A;. It is also clear that the associated sequence đ?&#x2018;Ś = (đ?&#x2018;Śđ?&#x2018;&#x2DC; ) is in đ?&#x2018;? and hence đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;?0 . Again, since đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; has AK (Lemma 2.1) and contains đ?&#x153;&#x2122;, by the mth partial sum of the series â&#x2C6;&#x17E; đ?&#x2018;&#x2DC;=0 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;&#x2DC; we have đ?&#x2018;&#x161; đ?&#x2018;&#x2DC;=0 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;&#x2DC;

=

đ?&#x2018;&#x161; đ?&#x2018;&#x2DC;=0

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2013;=1 đ?&#x2018;˘

đ?&#x2018;Łđ?&#x2018;&#x2013;

=

â&#x2C6;&#x17E; đ?&#x2018;&#x2DC;=0

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2013;=1 đ?&#x2018;˘

đ?&#x2018;Łđ?&#x2018;&#x2013;

đ?&#x2018;&#x2DC;

1

â&#x2C6;&#x2019;đ?&#x2018;Ł

1 đ?&#x2018;&#x2013;+1

+đ?&#x2018;˘

1 đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2DC;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Śđ?&#x2018;&#x2DC; ,

which becomes â&#x2C6;&#x17E; đ?&#x2018;&#x2DC;=0 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;&#x2DC;

đ?&#x2018;&#x2DC;

1

â&#x2C6;&#x2019;đ?&#x2018;Ł

1 đ?&#x2018;&#x2013;+1

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; + đ?&#x2018;˘

1 đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2DC;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Śđ?&#x2018;&#x2DC; , for đ?&#x2018;? > 1,

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Matrix Transformations on Some Difference Sequence Spaces

â&#x;š

đ??´đ?&#x2018;Ľ

đ?&#x2018;&#x2122;đ?&#x2018;?

â&#x2030;¤ supđ?&#x2018;&#x203A;

â&#x2030;¤ đ?&#x2018;Śđ?&#x2018;&#x2DC;

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2DC;=0 đ?&#x2018;˘ đ?&#x2018;&#x2DC;

đ?&#x2018;&#x2DC;

đ?&#x2018;&#x2122;đ?&#x2018;? đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x203A;

1 đ?&#x2018;Łđ?&#x2018;&#x2013;

â&#x2C6;&#x2019;

1 đ?&#x2018;Łđ?&#x2018;&#x2013;+1

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2DC;=0 đ?&#x2018;˘ đ?&#x2018;&#x2DC;

đ?&#x2018;&#x2DC;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Śđ?&#x2018;&#x2DC; +

1 đ?&#x2018;Łđ?&#x2018;&#x2013;

1

â&#x2C6;&#x2019;

đ?&#x2018;Łđ?&#x2018;&#x2013;+1

1 đ?&#x2018;˘ đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2DC;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Śđ?&#x2018;&#x2DC;

đ?&#x2018;? 1/đ?&#x2018;?

đ?&#x2018;? 1/đ?&#x2018;?

đ?&#x2018;? 1/đ?&#x2018;? đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 đ?&#x2018;&#x17D; đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC;=0 đ?&#x2018;˘ đ?&#x2018;Ł đ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC;

+

<â&#x2C6;&#x17E;

â&#x;š đ??´đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2122;đ?&#x2018;? and hence đ??´ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; â&#x2C6;ś đ?&#x2018;&#x2122;đ?&#x2018;? . Conversely, let đ??´ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; â&#x2C6;ś đ?&#x2018;&#x2122;đ?&#x2018;? , 1 < đ?&#x2018;? < â&#x2C6;&#x17E;. Then again by (2) and Lemma 1.1, đ??´đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; [đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; ]đ?&#x203A;˝ for all đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022; implying (ii) and (iii) for all đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; and đ?&#x2018;Ś â&#x2C6;&#x2C6; đ?&#x2018;&#x2122;đ?&#x2018;? . To prove (i), let the continuous linear functional đ?&#x2018;&#x201C;đ?&#x2018;&#x203A; (đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;) be defined on (đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; )â&#x2C6;&#x2014; , the continuous dual of đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; . Since the series â&#x2C6;&#x17E;đ?&#x2018;&#x2DC;=0 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;&#x2DC; converges for each đ?&#x2018;Ľ and for each đ?&#x2018;&#x203A;, then đ?&#x2018;&#x201C;đ??´đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; (đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; )â&#x2C6;&#x2014;; where đ?&#x2018;&#x201C;đ??´đ?&#x2018;&#x203A; đ?&#x2018;Ľ = â&#x2C6;&#x17E;đ?&#x2018;&#x2DC;=0 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;&#x2DC; â&#x2C6;&#x20AC; đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; . â&#x;š

đ?&#x2018;&#x201C;đ??´đ?&#x2018;&#x203A; = đ??´đ?&#x2018;&#x203A;

đ?&#x2018;&#x2122;đ?&#x2018;?

â&#x2C6;&#x17E; đ?&#x2018;&#x2DC;=0

=

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC;

1

đ?&#x2018;? đ?&#x2018;?

< â&#x2C6;&#x17E;, for all đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;,

with đ??´đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; [đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; ]đ?&#x203A;˝ . This means that the functional defined by the rows of A on đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; are pointwise bounded, and by the Banach-Steinhaus theorem these functional are uniformly bounded. Hence there exists a constant đ?&#x2018;&#x20AC; > 0, such that đ?&#x2018;&#x201C;đ??´đ?&#x2018;&#x203A; â&#x2030;¤ đ?&#x2018;&#x20AC;, â&#x2C6;&#x20AC; đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;, yielding (i). Theorem 3.2: đ??´ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; â&#x2C6;ś đ?&#x2018;?đ?&#x2018; if and only if conditions (ii) and (iii) of Theorem 3.1 hold, and đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x161;

(iv)

đ?&#x2018;&#x2DC;

đ?&#x2018;&#x161; đ?&#x2018;&#x203A;=1

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2013;=1 đ?&#x2018;˘

1 đ?&#x2018;Łđ?&#x2018;&#x2013;

đ?&#x2018;&#x2DC;

â&#x2C6;&#x2019;đ?&#x2018;Ł

1 đ?&#x2018;&#x2013;+1

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; + đ?&#x2018;˘

1 đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2DC;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; < â&#x2C6;&#x17E;.

Proof. Supposeđ??´ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; â&#x2C6;ś đ?&#x2018;?đ?&#x2018; . Then đ??´đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; [đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; ]đ?&#x203A;˝ for all đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;. Since đ?&#x2018;&#x2019;đ?&#x2018;&#x2DC; = đ?&#x203A;żđ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; , where đ?&#x203A;żđ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; = 1 (đ?&#x2018;&#x203A; = đ?&#x2018;&#x2DC;) and = 0 (đ?&#x2018;&#x203A; â&#x2030;  đ?&#x2018;&#x2DC;), belongs to đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; , the necessity of (ii) holds. Similarly by taking đ?&#x2018;Ľ = đ?&#x2018;&#x2019; = (1, 1, 1, â&#x20AC;Ś ) â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; we get (iii). We prove the necessity of (i) as follows: Suppose đ??´ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; â&#x2C6;ś đ?&#x2018;?đ?&#x2018;  . Then it implies đ?&#x2018;&#x161; đ?&#x2018;&#x203A;=1

đ??´đ?&#x2018;&#x; (đ?&#x2018;Ľ) < â&#x2C6;&#x17E;, đ?&#x2018;&#x161; = 1, 2, 3, â&#x20AC;Ś,

where, đ??´đ?&#x2018;&#x; đ?&#x2018;Ľ =

đ?&#x2018;&#x2DC;

đ?&#x2018;&#x17D;đ?&#x2018;&#x;đ?&#x2018;&#x2DC; (

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 đ?&#x2018;Ś đ?&#x2018;&#x2DC; đ?&#x2018;&#x2013;=0 đ?&#x2018;˘ đ?&#x2018;&#x2DC;

1 đ?&#x2018;Łđ?&#x2018;&#x2013;

â&#x2C6;&#x2019;đ?&#x2018;Ł

1 đ?&#x2018;&#x2013;+1

+đ?&#x2018;˘

đ?&#x2018;Śđ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2DC;

)

converges for each đ?&#x2018;&#x; whenever đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; , which follows by the Banach-Steinhaus theorem that đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x2DC; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; < â&#x2C6;&#x17E;, each đ?&#x2018;&#x;. Hence đ??´đ?&#x2018;&#x; defines an element of [đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; ]â&#x2C6;&#x2014; for each đ?&#x2018;&#x;. Now define đ?&#x2018;&#x17E;đ?&#x2018;&#x161; đ?&#x2018;Ľ =

đ?&#x2018;&#x161; đ?&#x2018;&#x203A;=1

đ??´đ?&#x2018;&#x; (đ?&#x2018;Ľ) ,

đ?&#x2018;&#x; = 1,2,3, â&#x20AC;Ś

đ?&#x2018;&#x17E;đ?&#x2018;&#x161; is subadditive. Moreover, đ??´đ?&#x2018;&#x; is a bounded linear functional on đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; implies each đ?&#x2018;&#x17E;đ?&#x2018;&#x161; is a sequence of continuous seminorms on đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; such that đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x161; đ?&#x2018;&#x17E;đ?&#x2018;&#x161; đ?&#x2018;Ľ = â&#x2C6;&#x17E;đ?&#x2018;&#x;=1 đ??´đ?&#x2018;&#x; (đ?&#x2018;Ľ) < â&#x2C6;&#x17E; for each đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; . Thus there exists a constant đ?&#x2018;&#x20AC; > 0 such that â&#x2C6;&#x17E; đ?&#x2018;&#x;=1

đ??´đ?&#x2018;&#x; (đ?&#x2018;Ľ) â&#x2030;¤ đ?&#x2018;&#x20AC; đ?&#x2018;Ľ

đ?&#x2018;? đ?&#x2018;˘,đ?&#x2018;Ł,â&#x2C6;&#x2020;

which implies (i). Sufficiency: Suppose (i) â&#x20AC;&#x201C; (iii) of the theorem hold. Then đ??´đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; [đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; ]đ?&#x203A;˝ . If đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; , it suffices to show that đ??´đ?&#x2018;&#x203A; (đ?&#x2018;Ľ) â&#x2C6;&#x2C6; đ?&#x2018;?đ?&#x2018; in the norm of the sequence space đ?&#x2018;?đ?&#x2018; . Now,

đ?&#x2018;&#x203A; đ?&#x2018;&#x2DC;=0 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;&#x2DC;

=

đ?&#x2018;&#x203A; đ?&#x2018;&#x2DC;=0

www.iosrjournals.org

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2013;=1 đ?&#x2018;˘

đ?&#x2018;&#x2DC;

1 đ?&#x2018;Łđ?&#x2018;&#x2013;

â&#x2C6;&#x2019;

1 đ?&#x2018;Łđ?&#x2018;&#x2013;+1

+

1 đ?&#x2018;˘ đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2DC;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; đ?&#x2018;Śđ?&#x2018;&#x2DC; 3 | Page


Matrix Transformations on Some Difference Sequence Spaces â&#x2030;¤ đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x203A;

đ?&#x2018;&#x203A; đ?&#x2018;&#x2DC;=0

â&#x2030;¤ đ?&#x2018;Śđ?&#x2018;&#x2DC; đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;&#x203A;

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2013;=0 đ?&#x2018;˘ đ?&#x2018;&#x2DC; â&#x2C6;&#x17E; đ?&#x2018;&#x2DC;=0

1 đ?&#x2018;Łđ?&#x2018;&#x2013;

â&#x2C6;&#x2019;đ?&#x2018;Ł

đ?&#x2018;&#x2DC;â&#x2C6;&#x2019;1 1 đ?&#x2018;&#x2013;=1 đ?&#x2018;˘

đ?&#x2018;Łđ?&#x2018;&#x2013;

đ?&#x2018;&#x2DC;

1

This implies đ??´đ?&#x2018;&#x203A; (đ?&#x2018;Ľ) â&#x2C6;&#x2C6; đ?&#x2018;?đ?&#x2018; or đ??´ â&#x2C6;&#x2C6; đ?&#x2018;? đ?&#x2018;˘, đ?&#x2018;Ł, â&#x2C6;&#x2020; â&#x2C6;ś đ?&#x2018;?đ?&#x2018;  .

1 đ?&#x2018;&#x2013;+1

â&#x2C6;&#x2019;

đ?&#x2018;&#x17D;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; + đ?&#x2018;˘ đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC;đ?&#x2018;Ł

đ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC;

1 đ?&#x2018;Łđ?&#x2018;&#x2013;+1

+

1 đ?&#x2018;˘ đ?&#x2018;&#x2DC; đ?&#x2018;Łđ?&#x2018;&#x2DC;

đ?&#x2018;Śđ?&#x2018;&#x2DC;

by (i)

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2DC; < â&#x2C6;&#x17E;, as đ?&#x2018;&#x203A; â&#x2020;&#x2019; â&#x2C6;&#x17E;.

â&#x2013;Ą

Concluding Remarks The generalization obtained here still admit improvement in the sense that the conditions obtained here may further be simplified resulting in less restrictions on the involved matrices. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Ahmad, Z. U., and Mursaleen, KĹ&#x2018;the-Teoplitz Duals of Some New Sequence Spaces and Their Matrix Transformations, Pub. DĂŠ Lâ&#x20AC;&#x2122;institut MathĂŠmatique Nouvelle sĂŠrie tome 42 (56), 1987, p. 57-61 Altay, B., and F. BaĹ&#x;ar, Some Paranormed Sequence Spaces of Non-absolute Type Derived by Weighted Mean, J. math. Anal. Appl. 319, (2006), p. 494-508 Altay B., and F. BaĹ&#x;ar, The Fine Spectrum and the Matrix Domain of the Difference Operator â&#x2C6;&#x2020; on the Sequence Space lp , 0 < đ?&#x2018;? < 1), Comm. Math. Anal. Vol. 2 (2), 2007, p. 1-11 BaĹ&#x;arir, M., and E. E. Kara, On Some Difference Sequence Spaces of Weighted Means and Compact Operators, Ann. Funct. Anal. 2, 2011, p. 114-129 Boos, J., Classical and Modern Methods in Summability, Oxford Sci. Pub., Oxford, 2000 Kizmaz, H., On Certain Sequence Spaces, Canad. Math. Bull. Vol. 24 (2), 1981, p. 169-176 Maddox, I. J., Spaces of Strongly Summable Sequences, Quart. J. Math. Oxford,18 (2), 1967, p. 345-355 Polat, H., and F. BaĹ&#x;ar, Some Euler Spaces of Difference Sequences of Order mâ&#x2C6;&#x2014;, Acta Mathematica Scienta, 2007, 27B (2), p. 254266 Polat, H., Vatan K. and Necip S., Difference Sequence Spaces Derived by Generalized Weighted Mean, App. Math. Lett. 24 (5), 2011, p. 608-614 Simons, S., The Sequences Spaces,l(pv ) and m(pv ), Proc. London Math. Soc. 15 (3), 1965, p. 422-436

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