COSMOS: Journal of Engineering & Technology A Refereed Research Journal Vol 4 / No 2 / Jul - Dec 201 4 ISSN: 2231 -4210 THE MATRIX TRANSFORMATION ON ORLICZ SPACE OF ENTIRE SEQUENCE

Minaxi Rani 1

Abstract First we show that the set E ={sk:k=1,2,3,…….} Is a determining set ΓM . The set of all finite matrices transforming ΓM . In to a FK-space Y. denoted by ( ΓM . : y). We Characteristics the Classes (ΓM : y) When

Y = ( c0) π, cπ, ГM, lπ, ls, ^π, hπ. In summary we have the following table:

(c0 ) (cπ) ГM ls, ^π hπ Г Necessary and sufficient conditions on the matrix are obtained.

Some of these results have appeared already. But the approach to obtain these results in the present paper is determining set for ГM. First, we investigate a determining set for ГM and the we characterize the classes of matrix transformations involving ГM and other known sequence spaces.

Keywords : Determining set, entire sequence, Hahn sequence.

Introduction Let = {all finite sequences } A complex sequence whose k th term is will be denoted We Φrequire the following . by {x k} or x .An Orlicz function is a function M: [0, ∞)

[o, ∞) which is continuous, non-decreasing and convex Let = ( k) be a sequence of positive numbers. with M(0) = 0 M(x)> 0, for x > 0 and M(x) ∞ as x = m = The BK-space of all bounded (complex) ∞. If convexity of Orlicz function M is replaced→ by M(x→ ∞ + y) ≤ M(x) + M(y); then this function is called modulus ℓsequences x = {x k}. function, defined and discussed by Ruckle and Maddox. c0 =The BK-space of all null sequence. An Orlicz function M can always be represented in the following integral form: M(x) = dt, where q c = The BK-space of all convergent sequence. known as the kernel of M, is right differentiable for t ≥ () In respect of c 0, c, we have ||x|| = sup (x) |x k| 0,q(0) = 0, q(t) >0 for t > 0,q is non-decreasing and q(t) ∞ ∞ as t ∞. ℓ → → Lindenstrauss and Tzafriri used the idea of Orlicz Where x={x k} c0 c ∞. function to construct Orlicz sequence space, ∈ ⊂ ⊂ ℓ 1/k A sequence x = (x k) is said to be analytic if Sup (k) |x k| < ∞ ∞. The vector space of all analytic sequences will be ∞ denoted by . A sequence x is called entire sequence || ℓ = ∈ : < , > 0. ∞ ∧ = 0. The of all entire / where w = {all complex sequences } limseqiIences→ | | will be de denoted by . h = The Hahn sequence space is the BK-space h ofΓ all sequences x = The space with the {x k} such that ℓ ∞ ∞

|| ∞ || = > : ≤ 1 , | − |. → lim = 0. becomes a which is called an Orlicz The norm on h is given by ||x|| = ∞ sequence space. ∑ | − | 1Asst. Professor, M. M. P. G. College, Fatehabad 17

denote the space of all those complex sequence {x k} M being a modulus function. In other words, such that ℓ is a null sequence. | | {x , x + x , x + x ,. . . ,x + x +. . + x +. .. } belongs to 1 1 2 1 3 1 2 k with the norm The space is a metric space with the metric ℓ ||x|| s = |x 1| + |x 1 +x 2| +... + |x 1 + x 2 +. . . +x k| +. . Γ x − y (, ) = x () Γ = x = x: ∈ Γ ρ π For all x = {x k} and y = {y k} in Γ . x Given a sequence x = {x k} its nth section is the sequence (n) ⋀ = x = x : ∈ ⋀ x = { x , x ….., x , .. 0, 0, ….} π 1 2 n S(k) = (0, 0, …., ), 1 in the kth place and -1 in are FK-Spaces with the metric the (k + 1)th place.±1,0,0 … … Γ ⋀ Lemma 1: x − y Let X be a FK-space and E is a determining set X. Let Y (, ) = : = 1, 2, 3 … () π be an FK-space and A is a matrix. Suppose that either X has AK or A is row finite. x h = x = x: ∈ h Then A (X:Y) if and only if π (1) ∈The columns of A belong to Y and ∞ (2) A[E] is a bounded subset of Y. Then h is a BK − space with norm |x| Given a sequence x = {x k} and infinite matrix A = (a n k ), n, k =1, 2, 3…. = − π π Then A = transform is the sequence y = (y ) n x Where y = (ℓ ) = = = (x ) ∈ w: ∈ m n π ∑ (, = x 1,2,3 … . )ℎ ∑ . () = = (x) ∈ w: ∈ π

x = = (x ) ∈ w:→ lim exists πare BK-spaces with the norm Definition 2: [Wilansky: 1984] In respect of m, (c), c Let X be a BK-space. Then D = D(X) = {x || = ∈ () Φ:(,0) ≤ 1 x Lemma 2: has AK where M is a modulus function. ℓ = = (x) ∈ w: ∈ ℓ π Proof: Γ is a BK-space with the norm ∞ Let x = {x k} ℓ || = ∑ ∈ Γ We call But then (), , ℓ,∧, hare rate spaces 3. || Definition 1: ∈ Γ. and hence The space consisting of all those sequences x in w such that (2.1) || || , = → 0 → for some arbitrarily fixed ( ≥ + 1) || ∞, is denoted → 0 by →∞ > 0 Γ. 18

|| || || || || || ≥ ≥ ≥ ⋯ . ≥ , = ( ) → 0 ≥ + 1 || → ∞, (2.1) = = || = 1,2,3,…,. || ⇒ → → ∞

⇒This Γ completes has AK. the proof. = , , , … , ,0,0… = 1,2,3,…,.

Proposition 1: ℎ ∈ = 1,2,3…. Let {S k : k = 1, 2, 3 …} be the set of all sequence in let E = {S k : k || || = − Φ= 1,each 2, 3 of …}then whose E non is a − determining zero terms set ± for 1. the space Proof: Step 1. Γ. || || + − + ⋯ Recall that is a metric space with the metric d(x, y) =

Γ || || + − . || = t 1s1 + t 2s2 + …+ t msm. () Let A be the absolutely convex hull of E. Let x A. So that Then x = with (1.1) ∈ ∑ t1 + t 2 + …+t m = || || − | | ≤ 1 ∈ || || Then d(x, 0) = = 0 () () ≤ | , 0 + | , 0 + ⋯ + ||( , 0). But ≤Hence 1 x A. (1.3) ( , 0) = 1 = 1,2,3,…,. Hence d(x, 0) by using (1.1). also x Thus D ∈ . ≤ ∑ || ≤ 1 ∈ Case 2. ⊂Let . y be x and let HenceΦ x

(1.2) Thus∈ A . D. || || || ⊂ ≥ ≥ ⋯ ≥ Step 2: Let x Express y as a member of A as in case 1. ∈ . Since E is invariant under permutation of the terms of its ⇒ x ∈ Φ and d(x, 0) ≤ 1. members, so is A. ⇒ x = x, x, … . x, 0, … and Hence x ∈ . || || || , , , (1.4)ℎ Therefore D ⊂ . in both cases D (1.3) sup ≤ 1 From (1.2) and (1.4) A = D. ⊂ . || … . , Consequently E is a determining set for Case 1. Suppose that This completes the proof. Γ . Proposition 2:

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An infinite matrix A = (a nk ) is in class | | (4.2) →lim = 0 = 1,2,3… ∈ (Γ: ()) ⇔ (2.1) lim = 0 →

( + + ⋯ + ) (4.3)(, , … , ) ℎ (2.2) < ∞ (, ) Proof: Proposition 5: Γ and ∀n, k. In Lemma 1. Take X = has AK- property take Y = An infinite matrix A = (a ) is in the class (c 0)k be an FK-space. Γ nk Further more is a determining set E (as in given proposition 2)Γ k A ∈ (Γ : ℓ ) ⇔ (5.1) | | ( Also A[E] = A(s ) = {(a n1 + a n2 + …)} Again by Lemma 1. A = 1,2, … ) (i) The Columns∈ (Γ of: (A belong) ) to and k (5.2) < ∞ (ii) A(s ) is a bounded subset (, ) ( ) Proposition 6: But condition (). (i) An infinite matrix A = (a nk ) is in the class ⇔ (ii) : = 1,2,3 … ∀. ( ⋯ ) A ∈ (Γ: ℓ) ⇔ (6.1) | | < ∞ () Hence we conclude⇔ that < ∞ (, ) Proposition 7: A ( ) An infinite matrix A = (a nk ) is in the class ∈ Γ: () ⇔ (2.1) (2.2) . The completes the proof. Omitting the proofs, we formulate the following results: A ∈ (Γ:∧) ⇔ (7.1) < ∞ (, ) Proposition 3:

An infinite matrix A = (a ) is in the class nk (7.2)( , , … , ) ℎ ∧ and ∀n, k. A ∈ (Γ : ) ⇔ (3.1) →lim ( = 1,2, … ) Proposition 8 :

( + + ⋯ + ) (3.2) < ∞ An infinite matrix A = (a nk ) is in the class (, ) Proposition 4:

An infinite matrix A = (a nk ) is in the class A ∈ (Γ : ℎ ) ⇔ (8.1) : A ∈ (Γ:Γ) = 1,2,3 … ℎ . | + + ⋯ + | ⇔ (4.1) < ∞ (, )

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International Journal of mathematics and mathematical Sciences, Vol.2004,(No.68) 3755- ( + + ⋯ + ) (8.2) 3764. () 3. Jurimae, E.1994. Properties of domains of mappings (, + , + ⋯ + ,) on rate spaces and spaces with speed, Acta et. − Commentationes Universitan‘s Tartuensis, 970: 53- < ∞ 64. 4. Lindenstrauss, .1. and Tzafriri,L.971. On Orlicz sequence spaces, israel J.Math.10:345-355 5. Maddox, 1.J.1986. Sequence spaces defined by a References modulus, Math. Proc. Cambridge PhilosSoc.100: 1. Albert Wilansky, 1984. Summability through 161-66. , North-Holland. Amsterdam, 6. Ruckle, W.I-1. 1973. FK spaces in which the 1984. sequence of coordinate vectors is bounded, Canad. 2. Chandrasekhara Rao, K and Subramanian, J.Math.25:973-978. N.2004.The Orlicz space of entire sequences,

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