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The Matrix Transformation on Orlicz Space of Entire Sequence

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The Matrix Transformation on Orlicz Space of Entire Sequence 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 sequence space. 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. 3 known as the kernel of M, is right differentiableͤ for t ≥ Ȅ ͥ(ͨ) 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 = ƥ ͬ ∈ ͫ: ȕ ͇ ʦ ʧ < ,͚ͣͦ ͙ͧͣ͡ > 0Ʃ. ∞ ∧ = 0. The vector space of all entire &Ͱͥ ͥ/& 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 norm {x k} such that ℓ ∞ ∞ |ͬ&| ∞ & &ͮͥ & ɳ|ͬ|ɳ = ͚͢͝ƥ > ͉: ȕ͇ʦ ʧ ≤ 1Ʃ , ȕ ͟ |ͬ − ͬ |͙͙͗ͣͪͦ͛ͧ͢. ͕͘͢ &→lim ͬ = 0. &Ͱͥ &Ͱͥ becomes a Banach space which is called an Orlicz The norm on h is given by ||x|| = ∞ sequence space. ∑&Ͱͥ ͟ |ͬ& − ͬ&ͮͥ| 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 u is a null sequence. | Ğ|Ğ {x , x + x , x + x ,. ,x + x +. + x +. .. } belongs to 3 1 1 2 1 3 1 2 k ʪ͇ ʦ ` ʧʫ 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 u Ι Ι Ğ ͧͩͤ x − y Ι ̾(ͬ, ͭ) = ͇ ƥɴ ɴ Ʃ x (͟) Γϐ = Ƥx = ʜxΙʝ: ƴ Ƹ ∈ Γƨ ρ πΙ For all x = {x k} and y = {y k} in ΁ Ι Γ . x Given a sequence x = {x k} its nth section is the sequence ϐ ʜ Ιʝ (n) ⋀ = Ƥx = x : ƴ ΙƸ ∈ ⋀ƨ 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 … … Γϐ ͕͘͢ ⋀ϐ u Lemma 1: Ğ ͧͩͤ xΙ − yΙ Let X be a FK-space and E is a determining set X. Let Y ̾(ͬ, ͭ) = ƥɴ ɴ : ͟ = 1, 2, 3 … Ʃ (͟) πΙ be an FK-space and A is a matrix. Suppose that either X has AK or A is row finite. xΙ hϐ = Ƥx = ʜxΙʝ: ƴ Ƹ ∈ hƨ Then A (X:Y) if and only if πΙ (1) ∈The columns of A belong to Y and ∞ (2) A[E] is a bounded subset of Y. Then hϐ is a BK − space with norm ɳ|x|ɳ Given a sequence x = {x k} and infinite matrix A = (a n k ), ͬ& ͬ&ͮͥ n, k =1, 2, 3…. = ȕ ͟ ɴ Ι − Ιͮͥɴ &Ͱͥ π π Then A = transform is the sequence y = (y ) n xΙ Ϧ _ _ Ι Where y = (ℓ ) = ͡ = Ƥͬ = (x ) ∈ w: ƴ ΙƸ ∈ mƨ n π Ϧ ∑&Ͱͥ ͕)& ͬ& (͢, ͟ = 1,2,3 … . )ͫℎ͙͙͙ͪͦ͢ ∑ ͕)& ͬ& ͙ͬͧͨͧ͝. xΙ (͗ͤ)_ = Ƥͬ = (xΙ) ∈ w: ƴ Ƹ ∈ ͗ͤƨ πΙ xΙ _ Ι ͗ = Ƥͬ = (x ) ∈ w:Μ→Ϧ lim ƴ ΙƸ existsƨ πare BK-spaces with the norm Definition 2: [Wilansky: 1984] ϐ ͤ ϐ ϐ In respect of m , (c ) , c Let X be a BK-space. Then D = D(X) = {x Φ© ͧͩͤ ɳ|ͬ|ɳ_ = ʺϐ©ʺ ∈ (͟) Φ:͘(ͬ,0) ≤ 1ʝ xΙ Lemma 2: has AK where M is a modulus function. ℓ_ = Ƥͬ = (xΙ) ∈ w: ƴ Ƹ ∈ ℓƨ πΙ Proof: Γ΁ is a BK-space with the norm ∞ Φ© Let x = {x k} _ &Ͱͥ ℓ ɳ|ͬ|ɳ = ∑ ʺϐ©ʺ ∈ Γ΁ We call u But then Ğ (͗ͤ)_, ͗_, ℓ_,∧ϐ, hϐare rate spaces ʞ3ʟ. |3Ğ| Definition 1: ʪ͇ ʦ ` ʧʫ ∈ Γ. and hence The space consisting of all those sequences x in w such u that (2.1) Ğ |)| ͧͩͤ |3Ğ| ͘Ƴͬ, ͬ Ʒ = ʪ͇ ʦ ` ʧʫ → 0 ͕ͧ ͢ → u for some arbitrarily fixed (͟ ≥ ͢ + 1) |3Ğ|Ğ ∞, ͇ ʦ ` isʧ denoted → 0 ͕ͧ by ͟ →∞ > 0 Γ΁. 18 u Ğ |ͬͥ| |ͬͦ| |ͬͧ| |ͬ(| |)| ͧͩͤ |ͬ&| ͇ ʦ ʧ ≥ ͇ ʦ ʧ ≥ ͇ ʦ ʧ ≥ ⋯ . ≥ ͇ ʦ ʧ ͘Ƴͬ, ͬ Ʒ = ʰ͇ ʬ ʭʱ → 0 ͕ͧ ͢ (͟ ≥ ͢ + 1) |3Ĝ| → ∞,͖ͭ ͩͧ͛͢͝(2.1) $ ͇ ʠ ` ʡ ͬ ͙͆ͨ ͥ = ͧ͛͢ Ƶ͇ ƴ Ƹƹ = |3Ĝ| ͚ͣͦ ͝ = 1,2,3,…,͡. |)| ⇒ ͬ → ͬ ͕ͧ ͢ → ∞ ͇ ʠ ` ʡ ΁ This⇒ Γ completes has AK. the proof. ͎͕͙͟ ͍% = ƣ ͥ, ͦ, ͧ, … , %,0,0…Ƨ͚ͣͦ͞ = 1,2,3,…,͡. Proposition 1: ͨℎ͙͢ ͍% ∈ ̿ ͚ͣͦ ͞ = 1,2,3…͡.̻ͧͣ͠ Let {S k : k = 1, 2, 3 …} be the set of all sequence in let E = {S k : k |ͬͥ| |ͬͦ| ͬ = Ƶ͇ ʦ ʧ − ͇ ʦ ʧƹ ͍ͥ Φ= 1,each 2, 3 of …}then whose E non is a − determining zero terms set ± for 1. the space Proof: Step 1. Γ΁. |ͬͦ| |ͬͧ| + Ƶ͇ ʦ ʧ − ͇ ʦ ʧƹ ͍ͦ + ⋯ Recall that is a metric space with the metric d(x, y) = Γ΁ |ͬ(| |ͬ(ͮͥ| u + Ƶ͇ ʦ ʧ − ͇ ʦ ʧƹ ͍(. ͧͩͤ |3Ğͯ4Ğ|Ğ ʪ͇ ʦ ` ʧʫ = 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) ∈ ( & &Ͱͥ & ∑ ͨ ͧ t1 + t 2 + …+t m = |3u| |3Ġ~u| ( ` ` & ͇ ʠ ʡ − ͇ ʠ ʡ & ȕ |ͨ | ≤ 1 ͕͘͢ ͧ ∈ ̿ &Ͱͥ |ͬͥ| |ͬ(ͮͥ| Then d(x, 0) = ͇ ʦ ʧ ͖͙͙͗ͣͩͧ ͇ ʦ ʧ = 0 (ͥ) (ͦ) ≤ |ͨ'ɳ͘Ƴͧ , 0Ʒ + |ͨͦɳ͘Ƴͧ , 0Ʒ + ⋯ + ( |ͨ(|͘(ͧ , 0). But Hence≤ 1 ͖ͭ x ͩͧ͛͢͝ A. (1.3) & ͘(ͧ , 0) = 1͚ͣͦ ͟ = 1,2,3,…,͡. Hence d(x, 0) by using (1.1). also x Thus D ∈ ( . ≤ ∑&Ͱͥ |ͨ&| ≤ 1 ∈ Case 2. ⊂Let ̻. y be x and let HenceΦ x (1.2) Thus∈ A ̾. D. |ͭͥ| |ͭͦ| |ͬ(| ⊂ ͇ ʦ ʧ ≥ ͇ ʦ ʧ ≥ ⋯ ≥ ͇ ʦ ʧ 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 = ʜxͥ, xͦ, … . xΛ, 0, … ʝand Hence x u u u u v w ∈ ̻. ˫ |ͬͥ| |ͬͦ| |ͬͧ| ˯ ͇ ʬ ʭ , ͇ ʬ ʭ , ͇ ʬ ʭ , ˮ ˮ (1.4)͎ℎͩͧ Therefore D ⊂ ̻. in both cases D (1.3) sup u ≤ 1 Ġ From (1.2) and (1.4) A = D. ⊂ ̻. ˬ |ͬ(| ˰ ˮ … . , ͇ ʬ ʭ ˮ ˮ ˮ Consequently E is a determining set for ˭ ˱ Case 1. Suppose that ΁ This completes the proof. Γ . Proposition 2: 19 An infinite matrix A = (a nk ) is in class u ġ |͕)& | ͕)& (4.2) )→Ϧlim ƶ͇ ʬ ʭƺ = 0͚ͣͦ ͟ = 1,2,3… ̻ ∈ (Γ΁: (͗ͤ)_) ⇔ (2.1) lim ƴ Ƹ = 0 )→Ϧ ) ͧͩͤ (͕)& + ͕)& + ⋯ + ͕)& ) (4.3)͘(͕)ͥ, ͕)ͦ, … , ͕)& )ͧ͝ ͖͙ͣͩ͘͘͢ ͚ͣͦ ͙͕͗ℎ (2.2) ɴ ɴ < ∞ (͢, ͟) ) 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 Ϧ proposition 2)Γ ΁ ΁ _ )& k A ∈ (Γ : ℓ ) ⇔ (5.1) ȕ|͕ | ͙͙͗ͣͪͦ͛ͧ͢ (͟ Also A[E] = A(s ) = {(a n1 + a n2 + …)} )Ͱͥ = 1,2, … ) Again by Lemma 1.
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