MATRIX TRANSFORMATIONS BETWEEN GENERALIZED WEIGHTED CESARO SEQUENCE SPACES

MATRIX TRANSFORMATIONS BETWEEN GENERALIZED

WEIGHTED CESARO SEQUENCE SPACES

Ado Balili , Ahmadu Kiltho and Zakawat U. Siddiqui

Department of Mathematics and Statistics, University of Maiduguri, Borno State, Nigeria

ABSTRACT

The main purpose of this paper is to characterize the classes of the matrices 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 ,ℓ∞𝜎 𝑎𝑛𝑑 (𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , 𝑐𝜎) where 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 denotes the generalized weighted

Cesaro sequence spaces, 𝑐𝜎 and ℓ_∞𝜎 denote the space of all bounded sequences all of whose 𝜎 -mean (or invariant -mean) are equal and the space of 𝜎-boundedness respectively.

KEYWORDS: Generalized weighted Cesaro Sequence Space, Paranormed Sequence Space, Cesaro Sequence Space, Matrix Transformations

Mathematical Subject Classification: 40A05, 40C05, 40D05

1. INTRODUCTION

A sequence space is defined to be a linear space of real or complex sequences. Throughout this

paper ℕ, ℝ 𝑎𝑛𝑑 ℂ denote the set of non negative integers, the set of real numbers and the set of

complex numbers respectively. Let 𝜔 be the space of all (real or complex) sequences and

ℓ∞ 𝑐 𝑎𝑛𝑑 𝑐0 are respectively the space of all bounded sequences, the space of all convergent

sequences and the space of null sequences.

International Research Journal of Mathematics, Engineering and IT

ISSN: (2349-0322) Impact Factor- 5.489, Volume 4, Issue 11, November 2017

Website- www.aarf.asia, Email : [email protected] , [email protected]

Let 𝑝 = (𝑝𝑘) be a bounded sequence of strictly positive real numbers with sup𝑘𝑝𝑘 =

𝐻 𝑎𝑛𝑑 𝑀 = max⁡(1, 𝐻). Then the linear space ℓ 𝑝 𝑎𝑛𝑑 ℓ_∞(𝑝) were defined by Maddox [1]( see

also [3,4,5]) as follows:

ℓ 𝑝 = 𝑥 = 𝑥_𝑘 ∈ 𝜔: |𝑥_𝑘 ∞

𝑘=1

|𝑝𝑘 _{< ∞ 𝑤𝑖𝑡𝑕 0 < 𝑝}

𝑘 ≤ 𝐻 < ∞

ℓ_∞ 𝑝 = 𝑥 = 𝑥_𝑘 ∈ 𝜔: sup 𝑘

|𝑥_𝑘|𝑝𝑘 _{< ∞}

Which are complete spaces paranormed by

𝑔₁ 𝑥 = [ |𝑥_{𝑘 𝑘}|𝑝𝑘_]1_{𝑀 𝑎𝑛𝑑 𝑔}

2 𝑥 = sup𝑘|𝑥𝑘| 𝑝𝑘

𝑀 _{𝑖𝑓𝑓 𝑖𝑛𝑓𝑝𝑘 > 0.}

The concept of Banach limit and almost convergence have been generalized to those of invariant

means (or 𝜎-mean) and 𝜎-convergence respectively.

Definition 1.1 (invariant means). Let 𝜎 be a mapping of ℕ into itself. A continuous linear functional 𝜙 on ℓ_∞ is said to be invariant mean or 𝜎-mean, if and only if

( i ) 𝜙 𝑥 ≥ 0, 𝑥 = 𝑥_𝑘 ≥ 0, ∀𝑛 ∈ℕ

( ii ) 𝜙 𝑒 = 1

( iii ) 𝜙 𝑥𝜎 (𝑛 ) = 𝜙 𝑥 , ∀𝑥 ∈ ℓ∞

Definition 1.2 (𝜎-convergence) A sequence 𝑥 = (𝑥_𝑘) ∈ ℓ_∞ is 𝜎-convergent if and only if

lim𝑚 →∞𝑡𝑚 ,𝑛𝜎 𝑥 = lim𝑛→∞

𝑥𝑛+𝑇′𝑥𝑛+⋯+𝑇𝑚𝑥𝑛

(𝑚 +1) = 𝐿, uniformly in n , L being the common value of

all 𝜎-mean at x, i.e 𝐿 = 𝜎-limitx. Thus , the space of 𝜎-convergent sequences is given by

𝑐𝜎 _{= 𝑥 = 𝑥}

𝑘 ∈ 𝜔: lim_𝑚 𝑡𝑚 ,𝑛𝜎 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦 𝑖𝑛 𝑛, 𝑥 ∈ ℓ∞

Definition 1.3 (𝜎-boundedness).The space of 𝜎-boundedness (ℓ∞𝜎 ) can be defined in the

following way:

ℓ_∞𝜎 = {𝑧 ∈ 𝜔: sup 𝑚 ,𝑛

|𝜓_{𝑚 ,𝑛}(𝑧)| < ∞}

Where

𝜓_{𝑚 ,𝑛} 𝑧 = 𝑡_{𝑚 ,𝑛} 𝑥 − 𝑡_{𝑚 −1,𝑛}(𝑥)

= 1

(𝑚 +1) 𝑇 𝑗_{𝑥 −} 1

𝑚 𝑚

𝑗 =0 𝑚 −1𝑗 =0 𝑇𝑗𝑥

= 1

𝑚 (𝑚 +1) 𝑗[𝑥𝜎𝑗(𝑛) 𝑚

𝑗 =1 − 𝑥𝜎𝑗 −1(𝑛)]

= 1

𝑚 (𝑚 +1) 𝑗[ 𝑧𝑖 𝑕𝑗 𝑖=𝑑𝑗 𝑚

𝑗 =1 ].

With 𝑑𝑗 = 𝜎𝑗 −1 𝑛 + 1 𝑎𝑛𝑑 𝑕𝑗 = 𝜎𝑗 𝑛 .

That is 𝜓_{𝑚 ,𝑛} 𝐴𝑧 = 𝑑𝑚𝑛𝐴 𝑥 − 𝑑𝑚 −1,𝑛𝐴(𝑥)

= 𝛼(𝑛, 𝑘, 𝑚)𝑧_{𝑘 𝑘}

Where 𝛼 𝑛, 𝑘, 𝑚 = 1

𝑚 (𝑚 +1) 𝑗[ 𝑎𝑖𝑘 𝑕𝑗 𝑖=𝑑𝑗 𝑚

𝑗 =1 ].

In 1970, Shiue [6], studied and discussed the Cesaro sequence spaces 𝑐𝑒𝑠_𝑝 and 𝑐𝑒𝑠_∞ which were

defined as

𝑐𝑒𝑠_𝑝 = 𝑥 = 𝑥_𝑘 ∈ 𝜔: 1 𝑛 ∞ 𝑛 =1 |𝑥_𝑘 𝑛 𝑘=1

|𝑝 < ∞ , 1 < 𝑝 < ∞.

𝑐𝑒𝑠_∞ = 𝑥 = 𝑥_𝑘 ∈ 𝜔: sup_𝑛 1

𝑛 |𝑥𝑘 𝑛

𝑘=1 < ∞ , 𝑝 = ∞.

With the finite norms

| 𝑥 |_𝑝 = (1 𝑛 ∞

𝑛=1 𝑛𝑘=1|𝑥𝑘|𝑝 ) 1_𝑝

, 1 ≤ 𝑝 < ∞. (1.1)

And

| 𝑥 |_∞ = sup_𝑛1

𝑛 |𝑥𝑘| 𝑛

Leibowitz [7] studied some properties of these spaces and showed that 𝑐𝑒𝑠_𝑝(1 < 𝑝 < ∞)and

𝑐𝑒𝑠_∞ are separable Banach spaces with the above norms.

In 1974, Lim [8] defined these spaces in different form as

𝑐𝑒𝑠𝑝 = 𝑥 = 𝑥𝑘 ∈ 𝜔: ( 1 2𝑟 ∞

𝑟=0 |𝑥𝑟 𝑘 )𝑝 < ∞ , 𝑓𝑜𝑟 1 < 𝑝 < ∞.

𝑐𝑒𝑠_∞ = 𝑥 = 𝑥_𝑘 ∈ 𝜔: sup 𝑟 ≥0

1

2𝑟 𝑥𝑘 < ∞ , 𝑓𝑜𝑟 𝑝 = ∞.

Where 𝑑𝑒𝑛𝑜𝑡𝑒𝑠𝑟 a sum over the range 2𝑟 ≤ 𝑘 < 2𝑟+1, and determined its dual spaces and

characterized some matrix classes.

In 1977, Lim [9] also extended the above space 𝑐𝑒𝑠𝑝 to the space

𝑐𝑒𝑠 𝑝 𝑓𝑜𝑟 𝑝 = 𝑝_𝑟 , 𝑤𝑖𝑡𝑕 𝑖𝑛𝑓𝑝𝑟 > 0, defined as

𝑐𝑒𝑠 𝑝 = {𝑥 = 𝑥_𝑘 ∈ 𝜔: (1 2𝑟 ∞

𝑟=0 𝑥𝑟 𝑘 )𝑝𝑟 < ∞ }, where 𝑑𝑒𝑛𝑜𝑡𝑒𝑠𝑟 a sum over the range

2𝑟 ≤ 𝑘 < 2𝑟+1. He showed that the space was paranormed space, paranormed by

𝑕 𝑥 = ( (1

2𝑟 ∞

𝑟=0 |𝑥𝑟 𝑘|)𝑝𝑟) 1

𝑀_{, where 𝑀 = max 1, 𝐻 𝑤𝑖𝑡𝑕 sup𝑟𝑝𝑟 = 𝐻 < ∞.}

In year 1979, Johnson and Mohapatra [10] defined the Cesaro sequence space 𝑐𝑒𝑠 𝑝, 𝑞 for

positive sequence of real numbers 𝑝𝑛 , 𝑞𝑛 𝑎𝑛𝑑 𝑄𝑛 = 𝑞1+ 𝑞2+ ⋯ + 𝑞𝑛 as

𝑐𝑒𝑠 𝑝, 𝑞 = {𝑥 = 𝑥_𝑘 ∈ 𝜔: (1 𝑄𝑛 ∞

𝑛=1 𝑛𝑘=1𝑞𝑘|𝑥𝑘|)𝑝𝑟 < ∞ } and studied some inclusion

relations.

In the year 1997, Khan and Rahman [11] extended the space ces (p) to ces (p, q) in different way

as

𝑐𝑒𝑠 𝑝, 𝑞 = 𝑥 = 𝑥𝑘 ∈ 𝜔: ( 1 𝑄₂𝑟 ∞

𝑛 =1

𝑞𝑘 𝑛

𝑘=1

𝑥𝑘 )𝑝𝑟 < ∞ ,

For 𝑝 = 𝑝𝑟 𝑤𝑖𝑡𝑕 𝑖𝑛𝑓𝑝𝑟 > 0, 𝑄2𝑟 = 𝑞2𝑟 + 𝑞2𝑟+1 + ⋯ + 𝑞2𝑟+1−1and 𝑑𝑒𝑛𝑜𝑡𝑒𝑠𝑟 a sum over

the range 2𝑟 _{≤ 𝑘 < 2}𝑟+1_{. They determined its Kothe-Toeplitz dual and characterized some}

Remark 1.1 ( i ) if 𝑞𝑛 = 1, for all n , then 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 reduced to 𝑐𝑒𝑠(𝑝) studied by Lim [ 9 ]

( ii ) if 𝑝𝑛 = 𝑝, for all n and 𝑞𝑛 = 1, for all n, then 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 reduced to 𝑐𝑒𝑠𝑝 studied byLim

[8].

( iii ) obviously ℓ 𝑝 ⊂ 𝑐𝑒𝑠 𝑝 ⊂ 𝑐𝑒𝑠 𝑝, 𝑞 , 𝑓𝑜𝑟 𝑝_𝑟 ≥ 1.

2. SOME BASIC DEFINITIONS AND LEMMAS

Definition 2.1 A linear topological space X over the field of real numbers ℝ is said to be a paranormed space if there is a sub additive function 𝑝: 𝑋 → ℝ such that 𝑝 𝜃 = 0, 𝑝 −𝑥 =

𝑝(𝑥) and scalar multiplication is continuous , that is 𝛼_𝑛 − 𝛼 → 0, 𝑎𝑛𝑑 𝑝 𝑥_𝑛 − 𝑥 →

0 𝑖𝑚𝑝𝑙𝑦 𝑝 𝛼𝑛𝑥𝑛 − 𝛼𝑥 → 0, 𝑎𝑠 𝑛 → ∞, for all 𝛼 ∈ℝ, 𝑎𝑛𝑑 𝑥 ∈ 𝑋, where 𝜃 is the zero vector in

the linear space X.

Definition 2.2 (see [2, 3]) Let X be a sequence space, we define the Kothe-Toeplitz dual and generalized Kothe-Toeplitz dual as follow:

𝑋𝛼 = {𝑎 = (𝑎𝑘) ∈ 𝜔: |𝑎𝑘 𝑘𝑥𝑘| < ∞, 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋} (2.1)

and

𝑋𝛽 = {𝑎 = (𝑎𝑘) ∈ 𝜔: 𝑎𝑘 𝑘𝑥𝑘 < ∞, 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋} (2.2)

Definition 2.3 ( FazlurRahman and Rezaulkarim [14]) If (𝑞_𝑛) is a bounded sequence of positive real numbers, then for 𝑝 = 𝑝_𝑟 𝑤𝑖𝑡𝑕 𝑖𝑛𝑓𝑝_𝑟 > 0, the cesaro weighted sequence space 𝑐𝑒𝑠(𝑝, 𝑞) is

defined by

𝑐𝑒𝑠 𝑝, 𝑞 = {𝑥 = (𝑥_𝑘) ∈ 𝜔: ( 1 𝑄_2𝑟 ∞

𝑟=0 𝑞𝑟 𝑘𝑥𝑘 )𝑝𝑟 < ∞}, where 𝑄2𝑟 = 𝑞2𝑟 + 𝑞2𝑟+1 + ⋯ +

𝑞₂𝑟+1₋₁and 𝑑𝑒𝑛𝑜𝑡𝑒𝑠_𝑟 a sum over the range 2𝑟 ≤ 𝑘 < 2𝑟+1.

Definition 2.4 (FazlurRahman and Rezaulkarim [15]) For 𝑠 ≥ 1 define

𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 = {𝑥 = (𝑥𝑘) ∈ 𝜔: (𝑄2𝑟 ∞

𝑟=0

)−𝑠( 1

𝑄₂𝑟 |𝑞𝑘𝑥𝑘 𝑟

Where (𝑞𝑘) is a bounded sequence of real numbers, 𝑝 = (𝑝𝑟) with 𝑖𝑛𝑓𝑝𝑟 > 0 𝑄2𝑟 = 𝑞2𝑟 +

𝑞₂𝑟+1 + ⋯ + 𝑞₂𝑟+1₋₁and 𝑑𝑒𝑛𝑜𝑡𝑒𝑠_𝑟 a sum over the range 2𝑟 ≤ 𝑘 < 2𝑟+1.

Throughout this paper the following well known inequalities (see [2] and [3]) will be used

For any positive integer 𝐸 > 1 and any two complex numbers a and b we have

|𝑎𝑏| ≤ 𝐸(|𝑎|𝑡_𝐸−𝑡 _{+ |𝑏|}𝑝 _(2.3)

Where 𝑝 > 1 𝑎𝑛𝑑 1 𝑝 +

1 𝑡 = 1

Lemma 2.1 (FazlurRahman and Rezaulkarim [15]).The generalized weighted Cesaro sequence space 𝑐𝑒𝑠(𝑝, 𝑞, 𝑠) is a paranormed space, paranormed by

𝑤 𝑥 = ( ∞𝑟=0(𝑄2𝑟)−𝑠( 1

𝑄2𝑟 |𝑞𝑟 𝑘𝑥𝑘|)

𝑝𝑟₎1_{𝑀 (2.4)}

Provided 𝐻 = sup_𝑟𝑝_𝑟 < ∞, 𝑎𝑛𝑑 𝑀 = max 1, 𝐻 .

Lemma 2.2 (Mursaleen and Aiyub [13], Theorem 4.3.1) Let 1 < 𝑝_𝑟 ≤ sup_𝑟𝑝_𝑟 < ∞. Then

𝐴 ∈ (𝑐𝑒𝑠 𝑝, 𝑞 , 𝑐𝜎_{) if and only if, there exists an integer 𝐸 > 1, such that for all n.}

( i ) 𝑈 𝐸 = sup_𝑚 (𝑄₂𝑟max⁡((|𝑡 𝑛 ,𝑘,𝑚 |) 𝑡𝑟

𝑞𝑟 ∞

𝑟=0 ). 𝐸−𝑡𝑟 < ∞ (2.5)

Where, 1

𝑝𝑟+ 1

𝑡𝑟 = 1, 𝑟 = 0,1,2, …, 𝑡 𝑛, 𝑘, 𝑚 = 1

𝑚 +1 𝑎(𝜎

𝑗 𝑚

𝑗 =0 𝑛 , 𝑘).

( ii ) 𝑎_𝑘 = (𝑎_𝑛𝑘)_𝑛=1∞ ∈ 𝑐𝜎, 𝑓𝑜𝑟 𝑒𝑎𝑐𝑕 𝑘 (2.6)

i.e lim𝑚 ∞𝑘=1𝑡 𝑛, 𝑘, 𝑚 = 𝜇𝑘 uniformly in n, k fixed.

In this case, the 𝜎-limit of Ax is

lim

𝑚 𝑡 𝑛, 𝑘, 𝑚 𝑥𝑘 = 𝜇𝑘𝑥𝑘 ∞

𝑘=1 ∞

𝑘=1

.

Lemma 2.3 (Aiyub [13], Theorem 4.3.2) Let 1 < 𝑝_𝑟 ≤ sup_𝑟𝑝_𝑟 < ∞. Then 𝐴 ∈ (𝑐𝑒𝑠 𝑝, 𝑞 , ℓ_∞𝜎 ₎

if and only if

Where E is an integer such that 𝐸 > 1 and 1

𝑝𝑟+ 1

𝑡𝑟 = 1, 𝑟 = 0,1,2, …

Lemma 2.4 ([15], Theorem 1): If 1 < 𝑝_𝑟 ≤ sup_𝑟𝑝_𝑟 ≤ 𝐻 < ∞, and 1

𝑝𝑟 + 1

𝑡𝑟 = 1, 𝑟 = 0,1,2, …

𝑐𝑒𝑠+ 𝑝, 𝑞, 𝑠 = [𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 ]𝛽 _{= 𝜇(𝑡, 𝑠),}

𝜇 𝑡, 𝑠 = {𝑎 = (𝑎_𝑘) ∈ 𝜔: ∞_𝑟=0(𝑄₂𝑟)𝑠 𝑡𝑟−1 (𝑄₂𝑟max_𝑟|𝑎𝑘 𝑞𝑘|)

𝑡𝑟_{. 𝐸}−𝑡𝑟 _{< ∞, 𝐸 > 1}}

Lemma 2.5 ([15], Theorem 2): Let 1 < 𝑝_𝑟 ≤ 𝐻 < ∞. Then the continuous dual 𝑐𝑒𝑠∗_{(𝑝, 𝑞, 𝑠) is}

isomorphic to 𝜇(𝑡, 𝑠), defined by (4.5.1)

Lemma 2.6 ([15], Theorem 3): Let 1 < 𝑝𝑟 ≤ sup𝑟𝑝𝑟 ≤ 𝐻 < ∞. Then 𝐴 ∈ (𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , ℓ∞) if

and only if there exists an integer 𝐸 > 1, such that 𝑈(𝐸, 𝑆) < ∞ where

𝑈 𝐸, 𝑆 = sup 𝑛

(𝑄2𝑟 ∞

𝑟=0

𝐴𝑟 𝑛 )𝑡𝑟(𝑄2𝑟)𝑠 𝑡𝑟−1 . 𝐸−𝑡𝑟, 𝑎𝑛𝑑 1 𝑝𝑟

+ 1

𝑡𝑟

= 1, 𝑟 = 0,1,2, …

𝐋𝐞𝐦𝐦𝐚 𝟐. 𝟕 15 , Theorem 4 Let 1 < 𝑝_𝑟 ≤ sup_𝑟𝑝_𝑟 ≤ 𝐻 < ∞. Then 𝐴 ∈ (𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , 𝑐) if

and only if

( i ) 𝑎_𝑛𝑘 → 𝛼_𝑘(𝑛 → ∞, 𝑘 𝑖𝑠 𝑓𝑖𝑥𝑒𝑑)

( ii ) There exists an integer 𝐸 > 1 such that 𝑈(𝐸, 𝑆) < ∞, where 𝑈 𝐸, 𝑆 as defined

in Lemma 2.6

Lemma 2.8 ([15], Corollary 1) Let 1 < 𝑝_𝑟 ≤ sup_𝑟𝑝_𝑟 ≤ 𝐻 < ∞ . Then 𝐴 ∈ 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , 𝑐₀ if and only if

( i ) 𝑎_𝑛𝑘 → 0, (𝑛 → ∞, 𝑘 𝑖𝑠 𝑓𝑖𝑥𝑒𝑑)

( ii ) There exists an integer 𝐸 > 1 such that 𝑈(𝐸, 𝑆) < ∞ where 𝑈(𝐸, 𝑆) as defined

in Lemma 2.6

.

3. MAIN RESULTS

In this section we characterize the matrix classes 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , ℓ∞𝜎 𝑎𝑛𝑑 (𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , 𝑐𝜎) as our

Theorem 3.1 Let 1 < 𝑝𝑟 ≤ sup𝑟𝑝𝑟 ≤ 𝐻 < ∞. Then 𝐴 ∈ (𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , 𝑐𝜎) if and only if, there

exists an integer 𝐵 > 1, such that for all n

( I ) 𝑈 𝐵 = sup𝑚 ∞_𝑟=0(𝑄2𝑟)𝑠 𝑡𝑟−1 (𝑄₂𝑟max_𝑟( 𝑡 𝑛,𝑘,𝑚 𝑞𝑘 )

𝑡𝑟_{. 𝐵}−𝑡𝑟 _{< ∞ (3.1)}

1 𝑝𝑟

+ 1

𝑡𝑟

= 1, 𝑟 = 0,1,2, …

( ii ) 𝑎_𝑘 = (𝑎_𝑛𝑘)_𝑛=1∞ ∈ 𝑐𝜎 for each k (3.2)

i.e lim𝑚 ∞𝑘=1𝑡 𝑛, 𝑘, 𝑚 = 𝜇𝑘, 𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦 𝑖𝑛 𝑛 𝑘 𝑓𝑖𝑥𝑒𝑑.

The 𝜎-limit of 𝐴𝑥 is

lim_𝑚 ∞_𝑘=1𝑡(𝑛, 𝑘, 𝑚)𝑥_𝑘 = ∞_𝑘=1𝜇_𝑘𝑥_𝑘.

Proof. Necessity, suppose that 𝐴 ∈ 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , 𝑐𝜎 _{. Now 𝑡(𝑛, 𝑘, 𝑚)𝑥} 𝑘 ∞

𝑘=1 exists for each m

and 𝑥 ∈ (𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , whence ((𝑡(𝑛, 𝑘, 𝑚))_𝑘 ∈ [𝑐𝑒𝑠(𝑝, 𝑞, 𝑠)]𝛽 for each m (see Lemma 3.1) for

the generalized Kothe-Toeplitz dual of ces (p, q, s).

By Lemma 3.2 therefore, it follows that each (𝑓_𝑚𝑛)_𝑚 defined by

𝑓𝑚𝑛 𝑥 = 𝑡𝑚𝑛𝜎 𝐴𝑥 , is an element of [𝑐𝑒𝑠(𝑝, 𝑞, 𝑠)]𝛽. Since ces(p,q,s) is complete and further

sup_𝑚|𝑡_𝑚𝑛𝜎 _{𝐴𝑥 | < ∞, on ces(p,q,s). Arguing with uniform boundedness principle, we have}

condition (3.1). Since 𝑒𝑘 = (0,0,0, … ,1,0,0, … ) ∈ 𝑐𝑒𝑠(𝑝, 𝑞, 𝑠), condition (3.2) also holds.

Sufficiency: suppose that the conditions (3.1) and (3.2) hold. For fix 𝑛 ∈ ℕ and for very integer,

𝑕 ≥ 1, we have

(𝑄₂𝑟 𝑕

𝑟=0 )𝑠 𝑡𝑟−1 𝑄2𝑟 max𝑟(𝑞𝑘−1 𝑡 𝑛, 𝑘, 𝑚 )𝑡𝑟. 𝐵−𝑡𝑟,

≤ sup 𝑚

(𝑄₂𝑟 ∞

𝑟=0

)𝑠 𝑡𝑟−1 _𝑄

2𝑟max_𝑟 (𝑞𝑘−1 𝑡 𝑛, 𝑘, 𝑚 )𝑡𝑟. 𝐵−𝑡𝑟

So let 𝑕, 𝑚 → ∞ together with condition (3.1) and (3.2)

lim

𝑕→∞𝑚 →∞lim (𝑄2𝑟 𝑕

𝑟=0

)𝑠 𝑡𝑟−1 _𝑄

≤ sup𝑚 ∞𝑟=0(𝑄2𝑟)𝑠 𝑡𝑟−1 𝑄2𝑟max𝑟(𝑞𝑘−1 𝑡 𝑛, 𝑘, 𝑚 )𝑡𝑟. 𝐵−𝑡𝑟

Therefore

∞_𝑟=0(𝑄₂𝑟)𝑠 𝑡𝑟−1 𝑄₂𝑟max_𝑟(𝑞_𝑘−1 𝑡 𝑛, 𝑘, 𝑚 )𝑡𝑟. 𝐵−𝑡𝑟

≤ sup𝑚 ∞𝑟=0(𝑄2𝑟)𝑠 𝑡𝑟−1 𝑄2𝑟max𝑟(𝑞𝑘−1 𝑡 𝑛, 𝑘, 𝑚 )𝑡𝑟. 𝐵−𝑡𝑟 < ∞ (3.3)

Hence 𝜇𝑘 ∈ [𝑐𝑒𝑠(𝑝, 𝑞, 𝑠)]𝛽. Also since (𝑡(𝑛, 𝑘, 𝑚))𝑘 ∈ [𝑐𝑒𝑠(𝑝, 𝑞, 𝑠)]𝛽

Therefore the series 𝑡(𝑛, 𝑘, 𝑚)𝑥_{𝑘 𝑘} 𝑎𝑛𝑑 𝜇_{𝑘 𝑘}𝑥_𝑘 converge for each m and

𝑥 ∈ 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 .

For given 𝜀 > 0 𝑎𝑛𝑑 𝑥 ∈ 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 𝑐𝑕𝑜𝑜𝑠𝑒 𝑕 𝑠𝑢𝑐𝑕 𝑡𝑕𝑎𝑡

( ∞𝑟=𝑕+1(𝑄2𝑟)−𝑠( 1

𝑄2𝑟 |𝑞𝑟 𝑘𝑥𝑘|)

𝑝𝑟₎1_{𝑀 < 𝜀}

Since (3.2) holds there exists 𝑚₀ such that

𝑕_𝑘=1(𝑡 𝑛, 𝑘, 𝑚 − 𝜇_𝑘 | < 𝜀, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 > 𝑚₀.

Since (3.1) also holds, it follows that

∞_𝑟=𝑕+1(𝑡 𝑛, 𝑘, 𝑚 − 𝜇_𝑘 | is arbitrary small.

Therefore,

lim𝑚 ∞𝑘=1𝑡(𝑛, 𝑘, 𝑚)𝑥𝑘 = ∞𝑘=1𝜇𝑘𝑥𝑘 𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦 𝑖𝑛 𝑛

This completes the proof of the Theorem 3.1.

Theorem 3.2 Let 1 < 𝑝_𝑟 ≤ sup_𝑟𝑝_𝑟 = 𝐻 < ∞. Then 𝐴 ∈ (𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , ℓ_∞𝜎_{) if and only if}

sup_𝑚𝑛 𝑕_𝑟=0(𝑄₂𝑟)𝑠 𝑡𝑟−1 𝑄₂𝑟max_𝑟(𝑞_𝑘−1 𝑡 𝑛, 𝑘, 𝑚 )𝑡𝑟. 𝐵−𝑡𝑟 < ∞ (3.4)

Where B is an integer such that 𝐵 > 1 𝑎𝑛𝑑 1 𝑝𝑟+

1

Proof: Necessary, Suppose that ∈ (𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , ℓ_∞𝜎 ) . Now ∞_𝑘=1𝛼(𝑛, 𝑘, 𝑚)𝑧_𝑘 exists for each 𝑚 𝑎𝑛𝑑 𝑧_𝑘 ∈ 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , whence 𝛼_𝑘(𝑛, 𝑘, 𝑚)_𝑘 ∈ 𝑐𝑒𝑠∗ _{𝑝, 𝑞, 𝑠 = [𝑐𝑒𝑠(𝑝, 𝑞, 𝑠)]}𝛽

for each m by lemma 3.2

Therefore, it follows that each (𝑓𝑚𝑛)𝑚 define by

𝑓𝑚𝑛 𝑥 = 𝜓𝑚 ,𝑛 𝐴𝑧 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 [𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 ]𝛽, since ces (p, q, s) is complete and

further sup𝑚 ,𝑛|𝜓𝑚 ,𝑛 𝐴𝑧 | < ∞, 𝑜𝑛 𝑐𝑒𝑠(𝑝, 𝑞, 𝑠), so by arguing with uniform boundedness

principle , we have condition of the Theorem 3.2.

Sufficiency. Suppose that the condition holds. For fix 𝑛 ∈ ℕ, and for every integer

𝑕 ≥ 1, we have

(𝑄₂𝑟 𝑕

𝑟=0

)𝑠 𝑡𝑟−1 _𝑄

2𝑟max_𝑟 (𝑞𝑘−1 𝑡 𝑛, 𝑘, 𝑚 )𝑡𝑟. 𝐵−𝑡𝑟

≤ sup 𝑚 ,𝑛

(𝑄₂𝑟 ∞

𝑟=0

)𝑠 𝑡𝑟−1 _𝑄

2𝑟max_𝑟 (𝑞𝑘−1 𝑡 𝑛, 𝑘, 𝑚 )𝑡𝑟. 𝐵−𝑡𝑟

So,

lim𝑕→∞ 𝑕𝑟=0(𝑄2𝑟)𝑠 𝑡𝑟−1 𝑄2𝑟max𝑟(𝑞𝑘−1 𝑡 𝑛, 𝑘, 𝑚 )𝑡𝑟. 𝐵−𝑡𝑟

≤ sup_{𝑚 ,𝑛} ∞_𝑟=0(𝑄₂𝑟)𝑠 𝑡𝑟−1 𝑄₂𝑟 max_𝑟(𝑞_𝑘−1 𝑡 𝑛, 𝑘, 𝑚 )𝑡𝑟. 𝐵−𝑡𝑟

< ∞

Hence 𝛼(𝑛, 𝑘, 𝑚) ∈ [𝑐𝑒𝑠(𝑝, 𝑞, 𝑠)]𝛽. Therefore the series 𝛼(𝑛, 𝑘, 𝑚)𝑧𝑘 𝑘 converges

for each m and 𝑧 ∈ (𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 . This complete the proof of Theorem 3.2.

4. CONCLUSION

Recently several authors defined and studied weighted Cesaro sequence space 𝑐𝑒𝑠(𝑝, 𝑞). In

addition several generalizations of the above sequence space were obtained such as 𝑐𝑒𝑠(𝑝, 𝑞, 𝑠)

and characterization of the classes of matrices

FazlurRahman et al.,[15], which are the generalization of Khan and Rahman [11]. In this paper

we were able to characterize the classes of the infinite

matrices 𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , ℓ∞𝜎 𝑎𝑛𝑑 (𝑐𝑒𝑠 𝑝, 𝑞, 𝑠 , 𝑐𝜎) as our main results. More so, there is room for

more characterizations.

ACKNOWLEDGEMENT

The authors thank the anonymous referees for their valuable suggestions which led to the

improvement of the paper.

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