SOME SPACES OF A-IDEAL CONVERGENT SEQUENCES DEFINED BY MUSIELAK-ORLICZ FUNCTION

Konuralp Journal of Mathematics Volume 4 No. 2 pp. 169–176 (2016) c KJM SOME SPACES OF A-IDEAL CONVERGENT SEQUENCES DEFINED BY MUSIELAK-ORLICZ FUNCTION SELMA ALTUNDAG AND MERVE ABAY Abstract. We introduce basic properties of some sequence spaces using ideal convergent and Musielak Orlicz function M = (Mk ). Including relations related to these spaces are investigated in this paper. 1. Introduction, Definitions and Notations Throughout this article w, c, c0 , l∞ , lp denote the spaces of all, convergent, null, bounded and p -absolutely summable sequences, where 1 ≤ p < ∞. Firstly, the notion of I -convergence was introduced by Kostryrko et all [1] and it is the generalization of statistical convergence. A = (ank ) be an infinite matrix of complex entries ank and x = (xk ) be a sequence ∞ P in w. If An (x) = ank xk converges for each, then we write n ∈ N. k=1 Definition 1.1. If X is a non-empty set then a family of sets I ⊆ 2X is ideal if and only if for each A, B ∈ I we have A ∪ B ∈ I and for each A ∈ I and each B ⊂ A we have B ∈ I.[1] Definition 1.2. A non-empty family of sets F ⊂ 2X is said to be a filter on X if and only if ∅ ∈ / F , for each A, B ∈ F we have A ∩ B ∈ F and for each A ∈ F and each B ⊃ A we have B ∈ F .[1] Definition 1.3. An ideal I 6= ∅ is called non-trivial if I 6= ∅ and X ∈ / I.[1] Definition 1.4. A non-trivial I ⊆ 2X is called admissible ideal if and only if {{x} : x ∈ X} ⊂ I.[1] Definition 1.5. A sequence x = (xn ) ∈ w is said to be I -convergent to L if there exists L ∈ C such that for all ε > 0 , the set {n ∈ N : |xn − L| ≥ ε} ∈ I. We say x, I − convergent to L and we write I − lim x = L. The number L is called I − limit of x.[2] 2010 Mathematics Subject Classification. 40A05, 46A45, 46E30. Key words and phrases. ideal; I-convergence; paranorm space; Musielak-Orlicz function. 169 170 SELMA ALTUNDAG AND MERVE ABAY Definition 1.6. An Orlicz function M is a function which is continuous, nondecreasing, and convex with M(0) = 0, for x > 0 and M(x) → ∞ as x → ∞. Lindenstrauss and Tzafriri [4] used the idea of Orlicz function to construct the sequence space ( )   ∞ X |xk | lM = x ∈ w : < ∞ for some ρ> 0 M ρ k=1 which is called an Orlicz sequence space. The space lM becomes a Banach space with the norm ( kxk = inf ρ>0: ∞ X k=1  M |xk | ρ )  ≤1 . The space lM is closely related to the space lp which is an Orlicz sequence space with M(x) = xp for 1 ≤ p < ∞. Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [5], Bhardwaj and Singh [6] and many others. It is well known that since M is a convex function and M(0) = 0 then M(tx) ≤ tM(x) for all t with 0 < t < 1. Dutta and Bas.ar [18] have recently introduced and stud0 ied the Orlicz sequence spaces lM (C, Λ) and hM (C, Λ) generated by Cesàro mean of order one associated with a fixed multiplier sequence of non-zero scalars. The readers may refer to [17] for relevant terminology and details on the algebraic and topological properties on sequence spaces. An Orlicz function M is said to satisfy ∆2 − condition for all values of u, if there exists constant K > 0 such that M (2u) ≤ KM (u) (u ≥ 0). The ∆2 − condition is equivalent to the inequality M(Lu) ≤ KLM(u) satisfying for all values of u and for L > 1 [7]. A sequence M = (Mk ) of Orlicz function is called a Musielak-Orlicz function see [8], [9]. The sequence N = (Nk ) defined by Nk (v) = sup {|v| u − (Mk ) : u ≥ 0} , k = 1, 2, ... is called the complementary function of a Musileak-Orlicz function M = (Mk ). For a given Musileak-Orlicz function M = (Mk ), the Musileak-Orlicz sequence space tM and its subspace hM are defined as follows: tM = {x ∈ ω : IM (cx) < ∞ for some c > 0} , hM = {x ∈ ω : IM (cx) < ∞ for all c > 0} , where IM is a convex modular defined by IM (x) = ∞ X Mk (xk ), x = (xk ) ∈ tM . k=1 We consider tM equipped with the Luxemburg norm     x kxk = inf ρ > 0 : IM ≤1 ρ or equipped with the Orlicz norm SOME SPACES OF A-IDEAL CONVERGENT SEQUENCES DEFINED BY... 171   1 kxk = inf (1 + IM (ρx)) : ρ > 0 . ρ The following inequality will be used throughout this paper. Let p = (pk ) be a sequence of positive  real numbers with 0 < h = inf pn ≤ pn ≤ H = sup pn < ∞ and let D = max 1, 2H−1 . Then for ak , bk ∈ C, the set of complex numbers for all k ∈ N, we have 0 p p p |ak + bk | k ≤ D {|ak | k + |bk | k } . n o p H Also, |a| k ≤ max 1, |a| for all a ∈ C. (1.1) The notion of paranormed space was introduced by Nakano [10] and Simons [11] and many others. Definition 1.7. Let X be a linear metric space. A function g : X → R is called paranorm if (1) g (x) ≥ 0, for all x ∈ X, (2) g (−x) = g (x), for all x ∈ X, (3) g (x + y) ≤ g (x) + g (y), for all x, y ∈ X, (4) if (λn ) be a sequence of scalars with λn → λ as n → ∞ and (xn ) is a sequence of vectors with g (xn − x) → 0 as n → ∞, then g (λn xn − λx) → 0 as n → ∞. Definition 1.8. A sequence space X is solid (or normal) if (αn xn ) ∈ X whenever (xn ) ∈ X for all sequences (αn ) of scalars with |αn | ≤ 1 for all n ∈ N. Definition 1.9. A sequence space X is said to be monotone if it contains the canonical preimages of its step spaces.[19] Lemma 1.1. If a sequence space X is solid, then X is monotone.[12] Definition 1.10. A sequence space X is sequence algebra if xy = (xn yn ) ∈ X whenever x = (xn ) , y = (yn ) ∈ X. We define the following sequence spaces in this article,   pk  |Ak (x) − L| cI (M, A, p) = x ∈ w : I − lim Mk =0 k ρ cI0 (M, A, p) =    pk |Ak (x)| x ∈ w : I − lim Mk =0 k ρ  l∞ (M, A, p) =  for some L and ρ > 0 ,  for some ρ > 0 ,    pk |Ak (x)| < ∞ for some ρ > 0 . x ∈ w : sup Mk ρ k Also we write mI (M, A, p) = cI (M, A, p) ∩ l∞ (M, A, p) mI0 (M, A, p) = cI0 (M, A, p) ∩ l∞ (M, A, p). If we take A = λ, these spaces are respectively reduced to the spaces cI0 (M, λ, p), cI (M, λ, p), l∞ (M, λ, p), mI0 (M, λ, p), mI (M, λ, p) defined by Mursaleen and Sharma [19]. If we take pk = 1 for all k, M(x) = M (x) and A = I, we get the spaces cI0 (M), cI (M), l∞ (M), mI0 (M), mI (M) which were studied by Tripathy and Hazarika [14]. Our aim is to define the paranormed space of ideal convergent sequence space with matrix transformation and Musielak-Orlicz function. 172 SELMA ALTUNDAG AND MERVE ABAY 2. Main Results Theorem 2.1. Let M = (Mk ) be a Musielak-Orlicz function and p = (pk ) be a bounded sequence of positive real numbers. Then, the spaces cI (M, A, p), cI0 (M, A, p), mI (M, A, p) and mI0 (M, A, p) are linear. Proof. Let x, y ∈ cI (M, A, p) and α, β be scalars. So, there exist positive numbers ρ1 , ρ2 and for given ε > 0, we have    pk  |Ak (x) − L1 | ε A1 = k ∈ N : M k ≥ ∈ I, ρ1 2D    pk  |Ak (x) − L2 | ε A2 = k ∈ N : M k ∈ I. ≥ ρ2 2D Let ρ3 = max {2 |α| ρ1 , 2 |β| ρ2 }. Since M = (Mk ) is nondecreasing and convex function, we can obtain       |Ak (x) − L1 | |Ak (y) − L2 | |Ak (αx + βy) − (αL1 + βL2 )| < Mk +Mk . Mk ρ3 ρ1 ρ2 So, we have  pk  pk  pk     |Ak (αx + βy) − (αL1 + βL2 (...truncated)