DIFFERENCE OPERATOR ON DOUBLE SEQUENCES WITH AN ORLICZ FUNCTION

DIFFERENCE OPERATOR ON DOUBLE SEQUENCES WITH AN ORLICZ FUNCTION

Moon Das

B.H. College, Howly, Barpeta, Assam-781316, India.

E-mail: [email protected]

Abstract. In this article we introduce some vector valued difference double sequence spaces defined by Orlicz function. We study some of their properties like symmetricity, completeness etc. and prove some inclusion results.

Key words. Orlicz function, difference space, completeness, seminorm, regular convergence, symmetric space.

AMS Classification No. 40A05; 40B05; 46E30.

1. Introduction

Throughout the article 2w(q), _{2 ∞}(q), 2c(q), 2c0(q), 2c^R(q), ₂c₀^R(q) denote the spaces of all, bounded, convergent in Pringsheim’s sense, null in Pringsheim’s sense, regularly convergent and regularly null double sequences, defined over a seminormed space (X, q), seminormed by q.

An Orlicz function M is a mapping M :[0, ∞) → [0, ∞) such that it is continuous, non-decreasing and convex with M(0) = 0, M(x) > 0 for x > 0 and M(x) → ∞, as x → ∞.

Throughout, a double sequence is denoted by A = <ank >, a double infinite array of elements ank ∈ X for all n, k ∈ N.

The initial works on double sequences are found in Bromwich [2]. Later on it is studied by Hardy [3], Moricz [7], Moricz and Rhoades [8], Tripathy [9], Basarir and Sonalcan [1], Tripathy and Sarma ([10], [11]) and many others. Hardy [3] introduced the notion of regular convergence for double sequences.

Definition. A double sequence space E is said to be symmetric if <a_nk > ∈ E implies

<a_π(n)π(k)> ∈ E , where π is a permutation of N.

We introduce the following difference double sequence spaces.

_{2 ∞}(M, ,∆ q) ={<a_nk > ∈ ₂w(q) :

,

sup ^nk

n k

M q a ρ

∆ < ∞, for some ρ > 0}.

2c(M, ,∆ q) = {<ank > ∈ 2w(q) : a^nk L 0

M q ρ

∆ −

→ , as n, k → ∞ , for some ρ > 0}, where ∆a_nk =a_nk −a_n+1,k−a_{n k, +1}+a_n+1,k+1, for all n, k ∈ N.

Also <a_nk > ∈ 2c^R (M, ,∆ q) i.e. regularly convergent if <a_nk > ∈ 2c(M, ,∆ q) and the following limits hold:

There exists L_k ∈ X, such that a^nk L^k 0

M q ρ

∆ −

→ , as n→ ∞, for some ρ > 0 and all k ∈ N.

There exists Jn ∈ X , such that a^nk Jⁿ 0

M q ρ

∆ −

→ , as k→ ∞ , for some ρ > 0 and all n ∈ N.

The definitions of 2c0(M, ,∆ q) and ₂c₀^R (M, ,∆ q) follow from the above definition on taking L = Lk = Jn = θ , for all n, k ∈ N .

3. Main Results

Theorem 3.1. The classes Z(M, ∆, q), where Z = ₂c c,_{2 0},₂c^R,_{2 0}c^R and _{2 ∞}are linear spaces.

Theorem 3.2. Let (X, q) be a complete seminormed space. Then the spaces Z(M, ∆, q), where Z = ₂c^R,_{2 0}c^R and _{2 ∞}are complete seminormed spaces seminormed by

f(<a_nk>) = ^{1 1}

,

inf 0 : sup ⁿ sup ^k sup ^nk 1

n k n k

a a a

M q M q M q

ρ ρ ρ ρ

> + + ∆ ≤

Proof. Let us consider the space _{2 ∞}(M, , )∆ q . It is simple to prove that _{2 ∞}(M, , )∆ q is a seminormed space. Let <a_nkⁱ > be a Cauchy sequence in_{2 ∞}(M, , )∆ q . Then for fixed x0, r > 0, we have

0

( _nkⁱ _nk^j )

f a a

rx

< − > < ε for all i, j ≥ m₀ , ( m₀ ∈ N).

Also for r > 0, choose ⁰ 1 2

M rx ≥ .

By the definition of the seminorm

^{1 1 1 1}

,

sup sup sup 1

i j i j i j

n n k k nk nk

n k n k

a a a a a a

M q M q M q

ρ ρ ρ

− − ∆ − ∆

+ + ≤

⁰ 2 M rx

≤ …. …. …. (1)

^{1 1 0}

( ) 2

i j

n n

i j

nk nk

a a rx

M q M

f a a

− ≤

< − > , ^{1 1 0}

( ) 2

i j

k k

i j

nk nk

a a rx

M M

f a a

− ≤

< − >

and ⁰

( ) 2

i j

nk nk

i j

nk nk

a a rx

M q M

f a a

∆ − ∆

< − > ≤

(

)

0

2 . 2

i j

n n

q a a rx

rx ε ε

− < = for all i j, ≥m₀.

(

)

0

2 . 2

i j

k k

q a a rx

rx ε ε

− < = for all i j, ≥m₀.

( )

0

2 . 2

i j

nk nk

q a a rx

rx ε ε

∆ − ∆ < = for all i j, ≥m₀.

Thus <aⁱ_n1> , <a₁ⁱ_k > and < ∆a_nkⁱ > are Cauchy sequences in X. Since X is complete so there exist a_n1, a_1k, y_nk∈X such that

_{1 1 1}

lim _nⁱ 1 , lim ⁱ_{k k} and lim _nkⁱ _nk.

i a an i a a i a y

→∞ = →∞ = →∞∆ =

From this it is clear that lim _nkⁱ .

i a X

→∞ ∈

Since M is continuous so taking j → ∞ in (1) we get

^{1 1 1 1}

,

sup sup sup 1

i i i

n n k k nk nk

n k n k

a a a a a a

M q M q M q

ρ ρ ρ

− − ∆ − ∆

+ + ≤

Taking infimum of such ρ’s, we get

1 1 1 1

,

inf : sup sup sup 1

i i i

n n k k nk nk

n k n k

a a a a a a

M q M q M q

ρ ε

ρ ρ ρ

− − ∆ − ∆

+ + ≤ <

for all i j, ≥m₀. Hence <a_nkⁱ −a_nk >∈_{2 ∞}(M, , )∆ q . Since _{2 ∞}(M, , )∆ q is linear, so

, 2 ( , , )

i i

nk nk nk

a a a _∞ M q

< − > < >∈ ∆ implies <a_nk > = <a_nk−a_nkⁱ +a_nkⁱ >∈_{2 ∞}(M, , )∆ q . Thus _{2 ∞}(M, , )∆ q is complete. The other cases can be proved using similar technique.

Proposition. 3.4. (i) Z(M, ∆, q) ⊆ _{2 ∞}(M, ∆, q) for Z = ₂c^R, ₂c₀^R. The inclusions are strict.

(ii) Z(M, q) ⊆ Y (M, ∆, q) for Z = ₂c, ₂c^R and Y = _{2 0}c , _{2 0}c^R respectively. The inclusions are strict.

Proof. (i) The first part is obvious. To show the inclusions are strict, consider the following example.

Example 3.2. Let X = C, M(x) = x and q(x) = |x|. Let the sequence <a_nk> be defined by a_nk = + , for n odd and all k ∈ N. n k

= 0, otherwise.

Then <a_nk >∈_{2 ∞}(M, , )∆ q but <a_nk >∉Z M( , , )∆ q for Z = ₂c^R, ₂c₀^R.

(ii) We prove 2c(M, q) ⊆ _{2 0}c (M, ∆, q). Let <a_nk >∈2c(M, q). Then for some ρ > 0,

a^nk L

M q ε

ρ

− < for all n≥n₀, k≥k₀, ( ,n k_{0 0}∈N).

Let r = 4ρ. Now

a^nk a^nk a^{n 1,k} a^{n k, 1} a^{n 1,k 1}

M q M q

r r

+ + + +

− − +

∆ =

= (a_nk L) ( a_n ^1,_k L) ( a_{n k}^{, 1} L) (a_n ^1,_k ¹ L)

M q r

+ + + +

− + − + + − + + −

≤ ^{1, , 1 1, 1}

4 4 4 4

n k n k n k

nk a L a L a L

a L

M q q q q

ρ ρ ρ ρ

+ − + − + + −

− + + +

≤1 4

ank L

M q ρ

− +1 ^{, 1}

4

an k L

M q ρ

+ −

+1 ^{, 1}

4

an k L

M q ρ

+ −

+1 ^{1, 1}

4

n k

a L

M q ρ

+ + −

< ε for all n≥n₀, k≥k₀, ( ,n k_{0 0}∈N) and for some r > 0.

Hence <a_nk >∈_{2 0}c (M, ∆, q). Thus ₂c(M, q) ⊆ _{2 0}c (M, ∆, q). Similarly it can be proved that 2c^R(M, q) ⊆ _{2 0}c^R(M, ∆, q).

To show the strict inclusions, consider the following example.

Example 3.3. Let X = C, M(x) = x and q(x) = |x|. Let the sequence <a_nk> be defined by a_nk = + , for all n, k ∈ N. n k

Clearly <a_nk >∈₂c M( , , )∆ q but<a_nk >∉₂c M q( , ).

Proposition 3.5. Let the Orlicz functions M, M₁, M2 satisfy the ∆2-conditions, then

(i) Z M( ₂, , )∆ q ⊆Z M( ₁, , )∆ q for Z = ₂c, _{2 0}c , ₂c^R, _{2 0}c^R if M x₁( )≤M₂( )x for all [0, )

x∈ ∞ .

(ii) Z M( ₁, , )∆ q ⊆Z M M( ₁, , )∆ q for Z = ₂c, _{2 0}c , ₂c^R, _{2 0}c^R.

(iii) Z M( ₁, , )∆ q ∩Z M( ₂, , )∆ q ⊆Z M( ₁+M₂, , )∆ q for Z = ₂c, _{2 0}c , ₂c^R, _{2 0}c^R. Proof. (i) The proof is obvious.

(ii) Consider Z = ₂c. Let <a_nk >∈₂c M( ₁, , )∆ q . Then for some ρ > 0,

₁ a^nk L 0 M q

ρ

∆ −

→ as n→ ∞, k→ ∞ .

Let _{nk 1} a^nk L

b M q

ρ

∆ −

= . Since b_nk → , there exists 0 n k₀, ₀∈N such that b_nk < for all 1 n≥n₀, k≥k₀.

Now by the remark, M b( _nk)≤b M_nk (1), for all n≥n₀, k≥k₀.

Thus ₁ a^nk L 0

M M q

ρ

∆ −

→ as n→ ∞, k→ ∞

and this implies that

( ₁) a^nk L 0

M M q

ρ

∆ −

→ as n→ ∞, k→ ∞ .

Hence <a_nk >∈(M M₁, , )∆ q . Similarly the result can be proved for the other spaces also.

(iii) Consider the case for Z = ₂c. Let <a_nk >∈₂c M( ₁, , )∆ q ∩₂c M( ₂, , )∆ q . Then for some ρ1, ρ2 > 0,

₁

1 2

ank L

M q ε

ρ

∆ −

< for all n≥n₀, k≥k₀, ( ,n k_{0 0}∈N).

₂

2 2

ank L

M q ε

ρ

∆ −

< for all n≥n₀^/, k≥k₀^/, ( ,n k₀^/ ₀^/∈N).

Let ρ = max {ρ1, ρ2 }, n₀^// =max{ ,n n_{0 0}^/}, k₀^// =max{ ,k k_{0 0}^/}. Now for n≥n₀^//, k ≥k₀^// and for some ρ > 0,

_{1 2 1 2}

1 2

( ) a^nk L a^nk L a^nk L

M M q M q M q

ρ ρ ρ

∆ − ∆ − ∆ −

+ ≤ +

2 2

ε ε ε

< + = .

Thus <a_nk >∈₂c M( ₁+M₂, , )∆ q . Hence the proof is complete. Similarly it can be proved for the other spaces also.

References

[1] Basarir, M. and Sonalcan, O.: On some double sequence spaces; J. Indian Acad.

Math. 21(2), (1999); 193-200.

[2] Bromwich T.J.IA: An Introduction to the Theory of Infinite Series; Macmillan and Co. Ltd. New York (1965).

[3] Hardy G. H.: On the convergence of certain multiple series; Proc. Camb. Phil. Soc.;

9 (1917), 86-95.

[4] Kamthan, P.K. and Gupta, M.: Sequence Spaces and Series: Marcel Dekker, 1980.

[5] Kizmaz, H: On certain sequence spaces; Canad. Math. Bull., 24 (1981), 169–176.

[6] Lindenstrauss, J. and Tzafriri, L.: On Orlicz sequence spaces: Israel J. Math. 10 (1971), 379-390.

[7] Moricz, F: Extension of the spaces c and c₀ from single to double sequences;

Acta. Math. Hungerica.; 57(1-2), (1991), 129-136.

[8] Moricz, F. and Rhoades B.E.: Almost convergence of double sequences and strong regularity of summability Matrices; Math. Proc. Camb. Phil. Soc.; 104 (1988), 283-294.

[9] Tripathy B.C.: A class of difference sequences related to p-normed space ^p; Demonstratio Math. 36 (4) (2003). 867-872.

[10] Tripathy B.C. and Sarma B.: Vector valued paranormed statistically convergent double sequence spaces; Mathematica Slovaca; 57(2) (2007), 179-188.

[11] Tripathy B.C. and Sarma B.: Statistically convergent double sequence spaces defined by Orlicz function; Soochow Journal of Mathematics; 32(2), (2005), 211-221.