The backward operator of double almost $\left(\lambda_{m}\mu_{n}\right)$ convergence in $\chi^{2}-$Riesz space defined by a Musielak-Orlicz function

(3s.) v. 37 3 (2019): 85–97. ISSN-00378712 in press doi:10.5269/bspm.v37i3.31971 Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm The Backward Operator of Double Almost (λm µn ) Convergence in χ2 −Riesz Space Defined By a Musielak-Orlicz Function N. Subramanian and A. Esi abstract: In this paper we introduce the backward operator is ∇ and study the notion backward operator of ∇− statistical convergence and backward operator of ∇− statistical Cauchy sequence using by almost (λm µn ) convergence in χ2 −Riesz space and also some inclusion theorems are discussed. Key Words: Aanalytic sequence, Musielak-Orlicz function, Double sequences, Chi sequence,Lambda, Riesz space, strongly, Statistical convergent. Contents 1 Introduction 85 2 Definitions and Preliminaries 86 3 The Backward operator of convergence of double almost (λm µn ) in χ2 Riesz space 89 4 Main Results 91 5 Competing Interests 96 6 Acknowledgement 96 1. Introduction Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write w2 for the set of all complex double sequences (xmn ), where m, n ∈ N, the set of positive integers. Then, w2 is a linear space under the coordinate wise addition and scalar multiplication. Some initial works on double sequence spaces is found in Tripathy [1] and Mursaleen [2] and Mursaleen and Edely [3,4], Subramanian and Misra [5], Pringsheim [6], Moricz and Rhoades [7], Robison [8], Savas et al. [9], Raj et al. [10], Francesco Tulone [11] and many others. Let Then the seP (xmn ) be a double sequence of real or complex numbers. P∞ ries ∞ x is called a double series. The double series m,n=1 mn m,n=1 xmn give one space is said to be convergent if and only if the double sequence (Smn )is convergent, where 2010 Mathematics Subject Classification: 40A05,40C05,40D05. Submitted May 17, 2016. Published April 08, 2017 85 Typeset by BSP style. M c Soc. Paran. de Mat. 86 N. Subramanian and A. Esi Smn = Pm,n i,j=1 xij (m, n = 1, 2, 3, ...) . A double sequence x = (xmn )is said to be double analytic if 1 supm,n |xmn | m+n < ∞. The vector space of all double analytic sequences are usually denoted by Λ2 . A sequence x = (xmn ) is called double entire sequence if 1 |xmn | m+n → 0 as m, n → ∞. The vector space of all double entire sequences are usually denoted by Γ2 . Let the set of sequences with this property be denoted by Λ2 and Γ2 is a metric space with the metric o n 1 (1.1) d(x, y) = supm,n |xmn − ymn | m+n : m, n : 1, 2, 3, ... , forall x = {xmn } and y = {ymn } in Γ2 . Let φ = {f inite sequences} . Consider a double sequence = (xmn ). The (m, n)th section x[m,n] of the sePxm,n [m,n] quence is defined by x = i,j=0 xij δ ij for all m, n ∈ N,   0 0 ...0 0 ... 0 0 ...0 0 ...   .      δ mn =  .  .    0 0 ...1 0 ... 0 0 ...0 0 ... with 1 in the (m, n)th position and zero otherwise. A double sequence x = (xmn ) is called double gai sequence if 1 ((m + n)! |xmn |) m+n → 0, as m, n → ∞. That is, |xmn | → 0. The double gai sequences will be denoted by χ2 . 2. Definitions and Preliminaries A double sequence x = (xmn ) has limit 0 (denoted by P − limx = 0) 1/m+n (i.e) ((m + n)! |xmn |) → 0 as m, n → ∞. (i.e) |xmn | → 0. We shall write more briefly as P − convergent to 0. An Orlicz function is a function M : [0, ∞) → [0, ∞) which is continuous, nondecreasing and convex with M (0) = 0, M (x) > 0, for x > 0 and M (x) → ∞ as x → ∞. If convexity of Orlicz function M is replaced by M (x + y) ≤ M (x) + M (y) , then this function is called modulus function. An Orlicz function M is said to satisfy ∆2 − condition for all values u, if there exists K > 0 such that M (2u) ≤ KM (u) , u ≥ 0. The Backward Operator of Double Almost (λm µn ) Convergence In χ2 −Riesz 87 Lemma 2.1. Let M be an Orlicz function which satisfies ∆2 − condition and let 0 < δ < 1. Then for each t ≥ δ, we have M (t) < Kδ−1 M (2) for some constant K > 0. A double sequence M = (Mmn ) of Orlicz function is called a Musielak-Orlicz function [see [12]]. A double sequence g = (gmn ) defined by gmn (v) = sup {|v| u − (Mmn ) (u) : u ≥ 0} , m, n = 1, 2, · · · is called the complementary function of a sequence of Musielak-Orlicz M . For a given sequence of Musielak-Orlicz function M, the Musielak-Orlicz sequence space tM is defined as follows n o tM = x ∈ w2 : IM (|xmn |)1/m+n → 0 as m, n, k → ∞ , where IM is a convex modular defined by P∞ P∞ 1/m+n IM (x) = m=1 n=1 Mmn (|xmn |) . Definition 2.2. A double sequence x = (xmn ) of real numbers is called almost P − convergent to a limit 0 if 1/m+n 1 Pr+p−1 Ps+q−1 ((m + n)! |xmn |) → 0. P − limp,q→∞ supr,s≥0 pq m=r n=s that is, the average value of (xmn ) taken over any rectangle {(m, n) : r ≤ m ≤ r + p − 1, s ≤ n ≤ s + q − 1} tends to 0 as both p and q to ∞, and this P − convergence h i is uniform in r and s. Let denote the set of sequences c2 . with this property as χ Definition 2.3. Let λ = (λm ) and µ = (µn ) be two non-decreasing sequences of positive real numbers such that each tending to ∞ and λm+1 ≤ λm + 1, λ1 = 1, µn+1 ≤ µn + 1, µ1 = 1. Let Im = [m − λm + 1, m] and In = [n − µn + 1, n] . For any set K ⊆ N × N, the number δ λ,µ (K) = limm,n→∞ 1 |{(i, j) : i ∈ Im , j ∈ In , (i, j) ∈ K}| , λ m µn is called the (λ, µ) − density of the set K provided the limit exists. Definition 2.4. A double sequence x = (xmn ) of numbers is said to be (λ, µ) − statistical convergent to a number ξ provided that for each ǫ > 0, limm,n→∞ λm1µ |{(i, j) : i ∈ Im , j ∈ In , |xmn − ξ| ≥ ǫ}| = 0, n that is, the set K (ǫ) = λm1µ |{(i, j) : i ∈ Im , j ∈ In , |xmn − ξ| ≥ ǫ}| has (λ, µ) − n density zero. In this case the number ξ is called the (λ, µ) − statistical limit of the sequence x = (xmn ) and we write St(λ,µ) limm,n→∞ = ξ. Definition 2.5. The double sequence θi,ℓ = {(mi , nℓ )} is called double lacunary if there exist three increasing sequences of integers such that 88 N. Subramanian and A. Esi m0 = 0, hi = mi − mi−1 → ∞ as i → ∞ and n0 = 0, hℓ = nℓ − nℓ−1 → ∞ as ℓ → ∞. Let mi,ℓ = mi nℓ , hi,ℓ = hi hℓ , and θ i,ℓ is determine by k ℓ , qℓ = nnℓ−1 . Ii,ℓ = {(m, n) : mi−1 < m < mi and nℓ−1 < n ≤ nℓ } , qk = mmk−1 Definition 2.6. Let M be an Orlicz function and P = (pmn ) be any factorable double sequence of strictly positive real numbers, we define the following sequence space:   χ2M ACλm µn , P =   " !#pmn 1/m+n   X 1 ((m + n)! |xm+r,n+s |) (xmn ) : P − lim M =0 , mn λm µn   ρ (m,n)∈Ir,s uniformly in r and s.     2 We shall denote χ2M ACλm µn , P as χ ACλm µn respectively when pmn = 1  for all m and n. If x is in χ2 ACλm µn , P , we shall say that x is almost (λm µn ) in χ2 strongly P −convergent with respect to the Orlicz function M   . Also note  if M (x) = x, pmn = 1 for all m, n and k then χ2M ACλm µn , P = χ2 ACλm µn , P , which are defined as follows:   χ2 ACλm µn , P =   " !#pmn 1/m+n   X 1 ((m + n)! |xm+r,n+s |) (xmn ) : P − lim M =0 , mn λm µn   ρ (m,n (...truncated)