Nagarajan Subramanian and Umakanta Misra THE NÖRLUND ORLICZ SPACE OF DOUBLE GAI SEQUENCES. 1. Introduction

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1 F A S C I C U L I A T H E A T I C I Nr Nagarajan Subramanian and Umakanta isra THE NÖRLUND ORLICZ SPACE OF DOUBLE GAI SEQUENCES Abstract Let χ denotes the space of all double gai sequences Let Λ denotes the space of all double analytic sequences This paper is devoted to a study of the general properties of Nörlund double Orlicz space of gai sequence space η ) χ and χ and Nörlund double Orlicz space of analytic sequence space η ) Λ and Λ Key words: analytic sequence, double sequence, Nörlund space, Orlicz space and gai space AS athematics Subject Classification: 40A05, 40C05, 40D05 1 Introduction Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively We write w for the set of all complex sequences x mn ), where m, n N, the set of positive integers Then, w is a linear space under the coordinate wise addition and scalar multiplication Some initial works on double sequence spaces are due to Bromwich [4] Later on, the double sequence spaces were studied by Hardy [5], oricz [9], oricz and Rhoades [10], Basarir and Solankan [], Tripathy [17], Turkmenoglu [19], and many others Let us define the following sets of double sequences: u t) := x mn ) w : sup x mn tmn <, m,n N C p t) := x mn ) w : p lim x mn l tmn = 1 for some l C, m,n C 0p t) := x mn ) w : p lim x mn tmn = 1, m,n L u t) := x mn ) w : x mn tmn <, m=1 n=1 C bp t) := C p t) u t) and C 0bp t) = C 0p t) u t) ;

2 13 Nagarajan Subramanian and Umakanta isra where t = t mn ) is the sequence of strictly positive reals t mn for all m, n N and p lim m,n denotes the limit in the Pringsheim s sense In the case t mn = 1 for all m, n N; u t), C p t), C 0p t), L u t), C bp t) and C 0bp t) reduce to the sets u, C p, C 0p, L u, C bp and C 0bp, respectively Now, we may summarize the knowledge given in some document related to the double sequence spaces Gökhan and Colak [1, ] have proved that u t) and C p t), C bp t) are complete paranormed spaces of double sequences and gave the α, β, γ duals of the spaces u t) and C bp t) Quite recently, in her PhD thesis, Zeltser [3] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences ursaleen and Edely [4] have recently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Cesàro summable double sequences Nextly, ursaleen [5] and ursaleen and Edely [6] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x = x jk ) into one whose core is a subset of the core of x ore recently, Altay and Basar [7] have defined the spaces BS, BS t), CS p, CS bp, CS r and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces u, u t), C p, C bp, C r and L u, respectively, and also examined some properties of those sequence spaces and determined the α duals of the spaces BS, BV, CS bp and the β ϑ) duals of the spaces CS bp and CS r of double series Quite recently Basar and Sever [8] have introduced the Banach space L q of double sequences corresponding to the well-known space l q of single sequences and examined some properties of the space L q Quite recently Subramanian and isra [9] have studied the space χ p, q, u) of double sequences and gave some inclusion relations We need the following inequality in the sequel of the paper For a, b 0 and 0 < p < 1, we have 1) a + b) p a p + b p The double series m,n=1 x mn is called convergent if and only if the double sequence s mn ) is convergent, where s mn = m,n i,j=1 x ijm, n N) see[1]) A sequence x = x mn ) is said to be double analytic if sup mn x mn 1/m+n < The vector space of all double analytic sequences will be denoted by Λ A sequence x = x mn ) is called double gai sequence if m + n)! x mn ) 1/m+n 0 as m, n The double gai sequences will be denoted by χ By φ, we denote the set of all finite sequences Consider a double sequence x = x ij ) The m, n) th section x [m,n] of the sequence is defined by x [m,n] = m,n i,j=1 x iji ij for all m, n N; where I ij

3 The Nörlund Orlicz space of denotes the double sequence whose only non zero term is i+j)! in the i, j)th place for each i, j N An FK-spaceor a metric space)x is said to have AK property if I mn ) 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 = x k ) x mn )m, n N) are also continuous Orlicz[13] used the idea of Orlicz function to construct the space L ) Lindenstrauss and Tzafriri [7] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space l contains a subspace isomorphic to l p 1 p < ) subsequently, different classes of sequence spaces were defined by Parashar and Choudhary [14], ursaleen et al [11], Bektas and Altin [3], Tripathy et al [18], Rao and Subramanian [15], and many others The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [6] Recalling [13] and [6], an Orlicz function is a function : [0, ) [0, ) which is continuous, non-decreasing, and convex with 0) = 0, x) > 0, for x > 0 and x) as x If convexity of Orlicz function is replaced by subadditivity of, then this function is called modulus function, defined by Nakano [1] and further discussed by Ruckle [16] and addox [8], and many others An Orlicz function is said to satisfy the condition for all values of u if there exists a constant K > 0 such that u) K u) u 0) The condition is equivalent to lu) Kl u), for all values of u and for l > 1 Lindenstrauss and Tzafriri [7] used the idea of Orlicz function to construct Orlicz sequence space ) xk l = x w : <, for some > 0 k=1 The space l with the norm x = inf > 0 : ) xk 1, k=1 becomes a Banach space which is called an Orlicz sequence space For t) = t p 1 p < ), the spaces l coincide with the classical sequence space l p If X is a sequence space, we give the following definitions: i) X = the continuous dual of X; ii) X α = a = a mn ) : m,n=1 a mn x mn <, for each x X ;

4 134 Nagarajan Subramanian and Umakanta isra iii) X β = a = a mn ) : m,n=1 a mn x mn is convegent, for each x X ; iv) X γ = a = a mn ) : sup N 1,N m,n=1 a mnx mn <, for each x X ; v) let X be an FK-space φ; then X f = fi mn ) : f X ; vi) X δ = a = a mn ) : sup mn a mn x mn 1/m+n <, for each x X ; X α, X β, X γ and X δ are called α or Köthe-Toeplitz) dual of X, β or generalized-köthe-toeplitz) dual of X, γ dual of X, δ dual of X respectively X α is defined by Gupta and Kamptan [0] It is clear that X α X β and X α X γ, but X α X γ does not hold, since the sequence of partial sums of a double convergent series need not to be bounded The notion of difference sequence spaces of single sequences was introduced by Kizmaz [30] as follows Z ) = x = x k ) w : x k ) Z for Z = c, c 0 and l, where x k = x k x k+1 for all k N Here c, c 0 and l denote the classes of convergent, null and bounded scalar valued single sequences respectively The above difference spaces are Banach spaces normed by x = x 1 + sup k 1 x k Later on the notion was further investigated by many others We now introduce the following difference double sequence spaces defined by Z ) = x = x mn ) w : x mn ) Z where Z = Λ, χ and x mn = x mn x mn+1 ) x m+1n x m+1n+1 ) = x mn x mn+1 x m+1n + x m+1n+1 for all m, n N Let P m, n ) m, n=0 be a sequence of non-negative real numbers with p 00 > 0 Consider the transformation y mn = 1 m n p ij i=0 j=0 m n p ij x m i, n j i=0 j=0 for m, n = 0, 1,, The set of all x mn ) for which y mn ) χ is called the Nörlund Orlicz space of double gai sequences The Nörlund Orlicz space of double gai sequences and is denoted by η χ ) Similarly the set of all xmn ) for which y mn ) Λ is called the Nörlund Orlicz space of double analytic sequences and is denoted by η Λ ) We write Pmn = p p mn, for m, n = 0, 1,,

5 The Nörlund Orlicz space of 135 All absolutely convex absorbent closed subset of locally convex Topological Vector Space X is called barrel X is called barreled space if each barrel is a neighbourhood of zero A locally convex Topological Vector Space X is said to be semi reflexive if each bounded closed set in X is σ X, X ) compact Hardy [31] gave regularity conditions for a Nörlund matrix Based on this Nörlund matrix transformation, in this paper the Nörlund Orlicz space η χ ) of double gai sequences is introduced Similar results hold for the Orlicz space of analytic sequences We have also examined as to whether the space χ is barrled space and semi reflexive space Definitions and preliminaries χ and Λ denote the Pringscheim s sense of double Orlicz space of gai sequences and Pringscheim s sense of double Orlicz space of bounded sequences respecctively The notion of a modulus function was introduced by Nakano [1] We recall that a modulus f is a function from [0, ) [0, ), such that a) f x) = 0 if and only if x = 0 b) f x + y) f x) + f y), for all x 0, y 0, c) f is increasing, d) f is continuous from the right at 0 Since f x) f y) f x y ), it follows from condition d) that f is continuous on [0, ) Define the sets: χ = x w m + n)! x mn ) 1/m+n : 0 as m, n for some > 0 and Λ = x w x mn 1/m+n : sup < for some > 0 m,n 1 The space Λ is a metric space with the metric xmn y mn 1/m+n d x, y) = inf > 0 : sup 1 m,n 1

6 136 Nagarajan Subramanian and Umakanta isra The space χ is a metric space with the metric m + n)! xmn y mn 1/m+n d x, y) = inf > 0 : sup 1 m,n 1 Proposition 1 η χ ) = χ 3 ain results Proof Let x = x mn ) η χ ) Then y χ so that for every ɛ > 0, we have a positive integer n 0 such that p 00 m + n)!x mn + + p mn 0 + 0)!x 00 < ɛ m+n for all m, n n 0 Take p 00 = 1; p 11 = = p mn = 0 We then have < ɛ m+n, m, n n 0 Therefore x = x mn ) χ Hence m+n)! xmn ) η χ ) χ On the other hand, let x = x mn ) χ But for any given ɛ > 0, there exists a positive integer n 0 such that m+n)! xmn < ɛ m+n, m, n n 0 We have m + n)! ymn p 00 m + n)!x mn + + p mn 0 + 0)!x 00 1 m + n)! xmn [p 00 + ] 0 + 0)! x00 + p mn 1 [ p00 ɛ m+n + + p mn ɛ 0+0] ɛm+n [p p mn ] ɛm+n = ɛ m+n, m, n n 0 Therefore y mn ) χ Consequently x η χ ) Hence 3) χ η χ ) From ) and 3) we obtain η χ ) = χ This completes the proof

7 that The Nörlund Orlicz space of 137 Proposition η Λ ) = Λ Proof Let x mn ) Λ Then there exists a positive constant T such m + n)! xmn T m+n for m, n = 0, 1,, m + n)! ymn p 00T m+n + + p mn T 0+0 T m+n [ p p ] mn T m+n T m+n [p p mn ] T m+n = T m+n, for m, n = 0, 1,, Hence y mn ) Λ But then x = x mn) η χ ) Consequently 4) Λ η Λ ) On the other hand let x mn ) η Λ ) Then ymn ) Λ Hence there exists a positive constant T such that ymn < T m+n for m, n = 0, 1,, This in turn implies that p 00 m + n)!x mn )!x 00 < T m+n Hence 1 and thus p00 m + n)!x mn + + p mn 0 + 0)!x 00 < T m+n p00 m + n)!x mn + + p mn 0 + 0)!x 00 < T m+n Take p 00 = 1; p 11 = = p mn = 0 Then it follows that = 1 and so m+n)! xmn < T m+n for all m, n Consequently x = x mn ) Λ Hence 5) η Λ ) Λ From 4) and 5) we get η Λ ) = Λ This completes the proof

8 138 Nagarajan Subramanian and Umakanta isra Proposition 3 χ Proof Let A = x χ : is not a barreled space m + n)! x mn ) 1 m+n 1 m + n, m, n Then A is an absolutely convex, closed absorbent in χ But A is not a neighbour hood of zero Hence χ is not barreled Proposition 4 χ is not semi reflexive Proof Let I mn) U be the closed unit ball in χ Clearly no subsequence of I mn) which weakly converges to any y χ Hence χ is not semi reflexive References [1] Apostol T, athematical Analysis, Addison-Wesley, London 1978 [] Basarir, Solancan O, On some double sequence spaces, J Indian Acad ath, 1)1999), [3] Bektas C, Altin Y, The sequence space l p, q, s) on seminormed spaces, Indian J Pure Appl ath, 344)003), [4] Bromwich TJI A, An Introduction to the Theory of Infinite Series, acmillan and CoLtd, New York 1965 [5] Hardy GH, On the convergence of certain multiple series, Proc Camb Phil Soc, ), [6] Krasnoselskii A, Rutickii YB, Convex Functions and Orlicz Spaces, Gorningen, Netherlands 1961 [7] Lindenstrauss J, Tzafriri L, On Orlicz sequence spaces, Israel J ath, ), [8] addox IJ, Sequence spaces defined by a modulus, ath Proc Cambridge Philos Soc, 1001)1986), [9] oricz F, Extentions of the spaces c and c 0 from single to double sequences, Acta ath Hung, 571-)1991), [10] oricz F, Rhoades BE, Almost convergence of double sequences and strong regularity of summability matrices, ath Proc Camb Phil Soc, ), [11] ursaleen, Khan A, Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstratio ath, Vol XXXII, 1999), [1] Nakano H, Concave modulars, J ath Soc Japan, 51953), 9-49 [13] Orlicz W, Über Raume L ), Bull Int Acad Polon Sci A, 1936), [14] Parashar SD, Choudhary B, Sequence spaces defined by Orlicz functions, Indian J Pure Appl ath, 54)1994), [15] Rao KC, Subramanian N, The Orlicz space of entire sequences, Int J ath ath Sci, 68004),

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