The Generalized Non-absolute type of sequence spaces

Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm (3s.) v. 34 2 (2016): 263–274. ISSN-00378712 in press doi:10.5269/bspm.v34i2.25674 The Generalized Non-absolute type of sequence spaces N. Subramanian, M.R.Bivin and N. Saivaraju abstract: In this paper we introduce the notion of λmn − χ2 and Λ2 sequences. iI(F ) h Further, we introduce the spaces χ2qλ f µ , k(d (x1 , 0) , d (x2 , 0) , · · · , d (xn−1 , 0))kp iI(F ) h , which are of non-absolute and Λ2qλ f µ , k(d (x1 , 0) , d (x2 , 0) , · · · , d (xn−1 , 0))kp type and we prove that these spaces are linearly isomorphic to the spaces χ2 and Λ2 , respectively. Moreover, we establish some inclusion relations between these spaces. Key Words: analytic sequence, double sequences, χ2 space, difference sequence space,Musielak - modulus function, p− metric space, Ideal; ideal convergent; fuzzy number; multiplier space; non-absolute type. Contents 1 Introduction 263 2 Notion of λmn − double chi and double analytic sequences 267 3 The spaces of λmn − double gai and double analytic sequences 269 4 Some Inclusion and Relations 270 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 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 Bromwich [1]. Later on, they were investigated by Hardy [2], Moricz [3], Moricz and Rhoades [4], Basarir and Solankan [5], Tripathy [6], Turkmenoglu [7], and many others. We procure the following sets of double sequences: n o t Mu (t) := (xmn ) ∈ w2 : supm,n∈N |xmn | mn < ∞ , n o t Cp (t) := (xmn ) ∈ w2 : p − limm,n→∞ |xmn − | mn = 1 f or some ∈ C , n o t C0p (t) := (xmn ) ∈ w2 : p − limm,n→∞ |xmn | mn = 1 , 2000 Mathematics Subject Classification: 40A05, 40C05, 46A45, 03E72, 46B20. 263 Typeset by BSP style. M c Soc. Paran. de Mat. 264 N. Subramanian, M.R.Bivin and N. Saivaraju n o P P∞ tmn Lu (t) := (xmn ) ∈ w2 : ∞ <∞ , m=1 n=1 |xmn | Cbp (t) := Cp (t) T Mu (t) and C0bp (t) = C0p (t) T Mu (t); where t = (tmn ) is the sequence of strictly positive reals tmn for all m, n ∈ N and p − limm,n→∞ denotes the limit in the Pringsheim’s sense. In the case tmn = 1 for all m, n ∈ N; Mu (t) , Cp (t) , C0p (t) , Lu (t) , Cbp (t) and C0bp (t) reduce to the sets Mu , Cp , C0p , Lu , Cbp and C0bp , respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. Gökhan and Colak [8,9] have proved that Mu (t) and Cp (t) , Cbp (t) are complete paranormed spaces of double sequences and gave the α−, β−, γ− duals of the spaces Mu (t) and Cbp (t) . Quite recently, in her PhD thesis, Zelter [10] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [11] and Tripathy have independently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Cesàro summable double sequences. Altay and Basar [12] have defined the spaces BS, BS (t) , CSp , CSbp , CSr and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces Mu , Mu (t) , Cp , Cbp , Cr and Lu , respectively, and also examined some properties of those sequence spaces and determined the α− duals of the spaces BS, BV, CSbp and the β (ϑ) − duals of the spaces CSbp and CSr of double series. Basar and Sever [13] have introduced the Banach space Lq of double sequences corresponding to the well-known space ℓq of single sequences and examined some properties of the space Lq . Quite recently Subramanian and Misra [14] have studied the space χ2M (p, q, u) of double sequences and gave some inclusion relations. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [15] as an extension of the definition of strongly Cesàro summable sequences. Connor [16] further extended this definition to a definition of strong A− summability with respect to a modulus where A = (an,k ) is a nonnegative regular matrix and established some connections between strong A− summability, strong A− summability with respect to a modulus, and A− statistical convergence. In [17] the notion of convergence of double sequences was presented by A. Pringsheim. Also, [18]- [19], and [20] the four dimensional P∞ in P ∞ matrix transformation (Ax)k,ℓ = m=1 n=1 amn kℓ xmn was studied extensively by Robison and Hamilton. We need the following inequality in the sequel of the paper. For a, b, ≥ 0 and 0 < p < 1, we have (a + b)p ≤ ap + bp (1.1) P∞ The double series m,n=1 xmn is called convergent if and only if the double seP quence (smn ) is convergent, where smn = m,n i,j=1 xij (m, n ∈ N). 1/m+n A sequence x = (xmn )is said to be double analytic if supmn |xmn | < ∞. The vector space of all double analytic sequences will be denoted by Λ2 . A sequence The Generalized Non-absolute type of sequence spaces 265 x = (xmn ) is called double gai sequence if ((m + n)! |xmn |)1/m+n → 0 as m, n → ∞. The double gai sequences will be denoted by χ2 . Let φ = {f inite sequences} . th [m,n] Consider a double sequence of the sequence P m,n x = (xij ). The (m, n) section x [m,n] is defined by x = i,j=0 xij ℑij for all m, n ∈ N ; where ℑij denotes the double th 1 sequence whose only non zero term is a (i+j)! in the (i, j) place for each i, j ∈ N. An FK-space(or a metric space)X is said to have AK property if (ℑmn ) is a Schauder basis for X. Or equivalently x[m,n] → x. An FDK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings x = (xk ) → (xmn )(m, n ∈ N) are also continuous. Let M and Φ are mutually complementary modulus functions. Then, we have: (i) For all u, y ≥ 0, uy ≤ M (u) + Φ (y) , (Y oung ′ s inequality)[See [21]] (1.2) (ii) For all u ≥ 0, uη (u) = M (u) + Φ (η (u)) . (1.3) (iii) For all u ≥ 0, and 0 < λ < 1, M (λu) ≤ λM (u) (1.4) Lindenstrauss and Tzafriri [22] used the idea of Orlicz function to construct Orlicz sequence space n   o P |xk | ℓM = x ∈ w : ∞ M < ∞, f or some ρ > 0 , k=1 ρ The space ℓM with the norm  n  o P∞ kxk = inf ρ > 0 : k=1 M |xρk | ≤ 1 , becomes a Banach space which is called an Orlicz sequence space. For M (t) = tp (1 ≤ p < ∞) , the spaces ℓM coincide with the classical sequence space ℓp . A sequence f = (fmn ) of modulus function is called a Musielak-modulus function. A sequence g = (gmn ) defined by gmn (v) = sup {|v| u − (fmn ) (u) : u ≥ 0} , m, n = 1, 2, · · · is called the complementary function of a Musielak-modulus function f . For a given Musielak modulus function f, the Musielak-modulus sequence space tf is defined as follows n o 1/m+n tf = x ∈ w2 : Mf (|xmn |) → 0 as m, n → ∞ , 26 (...truncated)