Vector Valued multiple of χ 2 over p metric sequence spaces defined by Musielak

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1 Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran ISSN: CJMS 6(2)(2017), Vector Valued multiple of χ 2 over p metric sequence spaces defined by Musielak Vandana 1,Deepmala 2, N Subramanian 3 and Lakshmi Narayan Mishra Department of Management Studies, Indian Institute of Technology, Madras, Chennai , Tamil Nadu, India vdrai1988@gmailcom 2 Mathematics Discipline, PDPM Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, PO: Khamaria, Jabalpur , Madhya Pradesh, India, dmrai23@gmailcom 3 Department of Mathematics, SASTRA University, Thanjavur , India, nsmaths@yahoocom 4 Department of Mathematics, Lovely Professional University, Jalandhar-Delhi GT Road, Phagwara, Punjab , India, L 1627 Awadh Puri Colony Beniganj, Phase -III, Opposite-Industrial Training Institute (ITI), Ayodhya Main Road Faizabad , Uttar Pradesh, India lakshminarayanmishra04@gmailcom, l n mishra@yahoocoin Abstract In this article, we define the vector valued multiple of χ 2 over p metric sequence spaces defined by Musielak and study some of their topological properties and some inclusion results Keywords: Analytic sequence, double sequences, χ 2 space,musielak - modulus function, p metric space, multiplier space 2000 Mathematics subject classification: 40A05; 40C05; Secondary 46A45; 03E72 1 Corresponding author: lakshminarayanmishra04@gmailcom Received: 6 February 2016 Revised: 21 September 2016 Accepted: 26 October

2 88 1 Introduction Consider w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectivelywe write w 2 for the set of all complex sequences (x mn ), where m, n N, the set of positive integers Then, w 2 is a linear space under the coordinate wise addition and scalar multiplication Throughout this article the space of regularly gai multiple sequence defined over a semi-normed space (X, q), semi-normed by q will be denoted by χ 2R mn (q) and Λ 2R mn (q) For X = C, the field of complex numbers, these spaces represent the corresponding scalar sequence spaces 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: M u (t) := (x mn ) w 2 : sup m,n N x mn t mn < }, C p (t) := (xmn ) w 2 : p lim m,n x mn l t mn = 1 for some l C }, C 0p (t) := (x mn ) w 2 : p lim m,n x mn t mn = 1 }, L u (t) := (x mn ) w 2 : m=1 n=1 x mn tmn < }, C bp (t) := C p (t) M u (t) and C 0bp (t) = C 0p (t) M u (t); 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; M u (t), C p (t), C 0p (t), L u (t), C bp (t) and C 0bp (t) reduce to the sets M 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 8,9 have proved that M 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 M u (t) and C bp (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), 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 M u, M 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 Basar and Sever 13

3 Vector Valued multiple of χ 2 over p metric 89 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 Misra 14 have studied the space χ 2 M (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 Cannor 16 further extended this definition to a definition of strong A summability with respect to a modulus where A = (a n,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, in 18-19, and 20 the four n=1 amn kl x mn was dimensional matrix transformation (Ax) k,l = m=1 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 a p + b p (11) 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 ij(m, n N) 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 2 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 2 Let ϕ = 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=0 x iji ij for all m, n N ; where I ij denotes 1 the double sequence whose only non zero term is a (i+j)! in the (i, j) th place for each i, j N An FK-space(or 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 t 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 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)see21 (12)

4 90 Vandana,Deepmala,N Subramanian,Lakshmi Narayan Mishra (ii) For all u 0, uη (u) = M (u) + Φ (η (u)) (13) (iii) For all u 0, and 0 < λ < 1, M (λu) λm (u) (14) Lindenstrauss and Tzafriri 22 used the idea of Orlicz function to construct Orlicz sequence space l M = x w : ( ) } k=1 M xk ρ <, for some ρ > 0, The space l M with the norm x = inf ρ > 0 : ( ) } k=1 M xk ρ 1, becomes a Banach space which is called an Orlicz sequence space For M (t) = t p (1 p < ), the spaces l M coincide with the classical sequence space l p A sequence f = (f mn ) of modulus function is called a Musielak-modulus function A sequence g = (g mn ) defined by g mn (v) = sup v u (f mn ) (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 t f is defined as follows } t f = x w 2 : M f ( x mn ) 1/m+n 0 as m, n, where M f is a convex modular defined by M f (x) = m=1 n=1 f mn ( x mn ) 1/m+n, x = (x mn ) t f We consider t f equipped with the Luxemburg metric ( ( )) } d (x, y) = sup mn inf m=1 n=1 f xmn 1/m+n mn 1 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 } ; (iii)x β = a = (a mn ) : m,n=1 a mn x mn is convegent, foreach x X } ; (iv)x γ } = a = (a mn ) : sup M,N mn 1 m,n=1 a mnx mn <, foreachx X ; (v)let X be an F K space ϕ; then X f = f(i mn ) : f X } ; mn

5 (vi)x δ = Vector Valued multiple of χ 2 over p metric 91 } a = (a mn ) : sup mn a mn x mn 1/m+n <, foreach x X ; X α, X β, X γ are called α (orköthe T oeplitz)dual of X, β (or generalized Köthe T oeplitz) dual ofx, γ dual of X, δ dual ofx respectivelyx α is defined by Gupta and Kampthan It is clear that X α X β and X α X γ, but X β 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 (for single sequences) was introduced by Kizmaz 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 difference sequence space bv p of the classical space l p is introduced and studied in the case 1 p by Başar and Altay and in the case 0 < p < 1 by Altay and Başar The spaces c ( ), c 0 ( ), l ( ) and bv p are Banach spaces normed by x = x 1 + sup k 1 x k and x bvp = ( k=1 x k p ) 1/p, (1 p < ) 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 2 : ( x mn ) Z } where Z = Λ 2, χ 2 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 2 Definition and Preliminaries Throughout this article a multiple sequence is denoted by; A = a m1 m 2 m rn 1 n 2 n s, a multiple infinite array of elements a m1 m 2 m rn 1 n 2 n s X for all m 1 m 2 m r n 1 n 2 n s N Let n N and X be a real vector space of dimension m, where n m A real valued function d p (x 1,, x n ) = (d 1 (x 1 ),, d n (x n )) p on X satisfying the following four conditions: (i) (d 1 (x 1 ),, d n (x n )) p = 0 if and and only if d 1 (x 1 ),, d n (x n ) are linearly dependent, (ii) (d 1 (x 1 ),, d n (x n )) p is invariant under permutation, (iii) (α m1 m 2 m rn 1 n 2 n s d 1 (x 1 ),, α m1 m 2 m rn 1 n 2 n s d n (x n )) p = α m1 m 2 m r n 1 n 2 n s (d 1 (x 1 ),, d n (x n )) p, α m1 m 2 m r n 1 n 2 n s R (iv) d p ((x 1, y 1 ), (x 2, y 2 ) (x n, y n )) = (d X (x 1, x 2, x n ) p + d Y (y 1, y 2, y n ) p ) 1/p for1 p < ; (or) (v) d ((x 1, y 1 ), (x 2, y 2 ), (x n, y n )) := sup d X (x 1, x 2, x n ), d Y (y 1, y 2, y n )}, for x 1, x 2, x n X, y 1, y 2, y n Y is called the p product metric of

6 92 Vandana,Deepmala,N Subramanian,Lakshmi Narayan Mishra the Cartesian product of n metric spaces is the p norm of the n-vector of the norms of the n subspaces A trivial example of p product metric of n metric space is the p norm space is X = R equipped with the following Euclidean metric in the product space is the p norm: (d 1 (x 1 ),, d n (x n )) E = sup ( det(d mn (x mn )) ) = d 11 (x 11 ) d 12 (x 12 ) d 1n (x 1n ) d 21 (x 21 ) d 22 (x 22 ) d 2n (x 1n ) sup d n1 (x n1 ) d n2 (x n2 ) d nn (x nn ) where x i = (x i1, x in ) R n for each i = 1, 2, n If every Cauchy sequence in X converges to some L X, then X is said to be complete with respect to the p metric Any complete p metric space is said to be p Banach metric space 21 Definition Let X be a linear metric space A function ρ : X R is called paranorm, if (1) ρ (x) 0, for all x X; (2) ρ ( x) = ρ (x), for all x X; (3) ρ (x + y) ρ (x) + ρ (y), for all x, y X; (4) If (σ mn ) is a sequence of scalars with σ mn σ as m, n and (x mn ) is a sequence of vectors with ρ (x mn x) 0 as m, n, then ρ (σ mn x mn σx) 0 as m, n A paranorm w for which ρ (x) = 0 implies x = 0 is called total paranorm and the pair (X, w) is called a total paranormed space It is well known that the metric of any linear metric space is given by some total paranorm (see 23, Theorem 1042, p183) 22 Definition A multiple sequence space E is said to be solid if α m1 m 2 m r n 1 n 2 n s a m1 m 2 m r n 1 n 2 n s E whenever a m1 m 2 m r n 1 n 2 n s E for all multiple sequences α m1 m 2 m r n 1 n 2 n s of scalars with α m1 m 2 m r n 1 n 2 n s 1 for all m 1 m 2 m r n 1 n 2 n s N 23 Definition A multiple sequence space E is said to be symmetric if a m1 m 2 m rn 1 n 2 n s a m1 m 2 m rn 1 n 2 n s E, implies a π(m1 m 2 m r n 1 n 2 n s ) E, where π (m 1 m 2 m r n 1 n 2 n s ) E are permutations of (M N) (M N) 24 Definition A multiple sequence space E is said to be monotone if it contains the canonical pre-images of all its step spaces

7 Vector Valued multiple of χ 2 over p metric Defintion A multiple sequence space E is said to be convergence free if b m1 m 2 m r n 1 n 2 n s E, whenever a m1 m 2 m r n 1 n 2 n s E and b m1 m 2 m r n 1 n 2 n s = 0 whenver a m1 m 2 m r n 1 n 2 n s = 0 26 Remark A sequence space E is solid implies E is monotone Let f be an Musielak modulus function Now we introduce the following multiple sequence spaces: Λ 2fq mn, (d (x 1 ), d (x 2 ),, d (x n 1 )) p = a m1 m 2 m r n 1 n 2 n s wmn 2q : sup m 1 ( m 2 m r n 1 n 2 n s )) (η uv (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) p <, where η uv (x) = a m1 m 2 m r n 1 n 2 n s (1/m 1m 2 m r)+n 1 n 2 n s χ 2fq mn, (d (x 1 ), d (x 2 ),, d (x n 1 )) p = a m1 m 2 m r n 1 n 2 n s wmn 2q : ( )) 0; as m 1 m 2 m r n 1 n 2 n s where µ uv (x) ((m 1 m 2 m r + n 1 n 2 n s )! a m1m 2 m rn 1n 2 n s ) (1/m 1m 2 m r )+n 1 n 2 n s A = a m1 m 2 m rn 1 n 2 n s χ 2Rf mn (q), (ie), regularly gai if a m1 m 2 m rn 1 n 2 n s χ 2f mn (q) and the following limit hold: ( )) 0; as m 1 n 1 and ( m 2 m r, n 2 n s N )) 0; as m 2 n 2 and m 3 m r, n 3 n s N ( )) 0 as m r n s and m 1 m r 1, n 1 n s 1 N 27 Remark The space χ 2Rf mn (q) has the following definition too χ 2fq mn, (d (x 1 ), d (x 2 ),, d (x n 1 )) p = a m1 m 2 m r n 1 n 2 n s w 2q f ( q )) 0; mn : as max m 1 m 2 m r n 1 n 2 n s } χ 2Bf mn (q) = χ 2f mn (q) Λ 2f mn (q) 3 Main result Theorem 31 χ 2Rf mn (q), χ 2Bf mn (q) and Λ 2f mn (q) of multiple sequences are linear spaces

8 94 Vandana,Deepmala,N Subramanian,Lakshmi Narayan Mishra Proof: We have to use linearity condition and then prove to the statement Hence it is trivial Therefore omit the proof Theorem 32 Let f = (f mn ) be a multiple sequence of Musielakmodulus functions Then then space χ 2Rf is a paranormed space with respect to the paranorm defined by g ( (x) = ( inf )))} 1 Proof: Clearly g (x) 0 for x = (a m1 m 2 m r,n 1 n 2 n s ) χ 2Rf Since f (0) = 0, we get g (0) = 0 Conversely, ( ( suppose that g (x) = 0, then )))} inf 1 = 0 Suppose that µ uv (x) 0 for each u, v N Then µ uv (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) p It follows that ( ( ))) which is a ( contradiction ( Therefore µ uv (x) = 0 Let ))) 1 and ( ( ))) ( µ uv (y), (d (x 1 ), d (x 2 ),, d (x n 1 )) p 1 Then by using Minkowski s inequality, we have ( ( ))) ( µ uv (x + y), (d (x 1 ), d (x 2 ),, d (x n 1 )) p ( ( ))) + ( ( ))) ( µ uv (y), (d (x 1 ), d (x 2 ),, d (x n 1 )) p ( So we( have g (x + y) = inf ))) } ( µ uv (x + y), (d (x 1 ), d (x 2 ),, d (x n 1 )) p 1 ( ( ))) } inf 1 + ( ( ))) } inf ( µ uv (y), (d (x 1 ), d (x 2 ),, d (x n 1 )) p 1 Therefore, g (x + y) g (x) + g (y) Finally, to prove that the scalar multiplication is continuous Let λ m1 m 2 m r n 1 n 2 n s be any complex number By definition, g (λ m1 m 2 m r n 1 n 2 n s x) = inf

9 Vector Valued multiple of χ 2 over p metric 95 ( ( ))) ( µ uv (λ m1 m 2 m r n 1 n 2 n s x), (d (x 1 ), d (x 2 ),, d (x n 1 )) p Then g (λ m1 m 2 m rn 1 n 2 n s x) = inf ( (( λ m1 m 2 m r n 1 n 2 n s t) : f } 1 ( ))) } q 1 where t = 1 λ Since λ max (1, λ suppuv ), we have g (λ m1 m 2 m r n 1 n 2 n s x) max (1, λ supp uv ( ( ) ))) inf t : ( µ uv (λ m1 m 2 m r n 1 n 2 n s x), (d (x 1 ), d (x 2 ),, d (x n 1 )) p This completes the proof Theorem 33 The space χ 2Rf is not symmetric Proof: Let a m1 m 2 m r,n 1 n 2 n s χ 2Rf Then for a given ϵ > 0 there exists a positive integers g 1, g 2, g g+1 h 1, h 2 h h+1 such that ( )) < ϵ for all m 1 n 1 > g 1 h 1 for all m 2 m r, n 2 n s N ( )) < ϵ for all m 2 n 2 > g 2 h 2 for all m 3 m r, n 3 n s N } 1 ( )) < ϵ for all m r n s > g r h s for all m 1 m r 1, n 1 n s 1 N ( )) < ϵ for all m 1 > g r+1, m 2 > g r+1,, m r > g r+1, n 1 > h s+1, n 2 > h s+1, n s > h s+1 N Let g 0 h 0 = max g 1, g 2, g r, g r+1, h 1, h 2 h r, h r+1 } Let b m1 m 2 m r,n 1 n 2 n s be a rearrangement of a m1 m 2 m r,n 1 n 2 n s Then we have a i1 i 2 i r,j 1 j 2 j s = b mi1 m i2 m ir,n j1 n j2 n js for all i 1 i 2 i r, j 1 j 2 j s N Let g r+2 h r+2 = max m 11 m 21 m r1, m r+11, m (r0 ) 1 n 11 n 21 n s1, n s+11 m 1r m 2r m rr, m r+1r, m (r0 ) r n 1s n 2s n ss, n s+1s Then we have

10 96 Vandana,Deepmala,N Subramanian,Lakshmi Narayan Mishra ( )) < ϵ for all m 1 n 1 > g r+2 h s+2 m r n s > g r+2 h s+2 Thus b mi1 m i2 m i r,n j 1 n j2 n j χ 2Rf s Hence χ 2Rf is a symmetric space Theorem 34 The spaces χ 2Rf and χ 2Bf are solid Proof: The spaces χ 2Rf and χ 2Bf are solid follows the following ( inequality )) (α m1 m 2 m r,n 1 n 2 n s µ uv (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) p ( )) for all m 1 m 2 m r, n 1 n 2 n s N and scalars α m1 m 2 m r,n 1 n 2 n s with α m1 m 2 m r,n 1 n 2 n s 1 for all m 1 m 2 m r, n 1 n 2 n s N Theorem 35 The spaces χ 2Rf and χ 2Bf are monotone Proof:The proof follows from the Remark 26 and Theorem 34 Theorem 36 Let f 1 and f 2 be multiple sequence of Musielak modulus functions Then we have (1) χ 2Rf 1 mn, (d (x 1 ), d (x 2 ),, d (x n 1 )) p χ 2Rf 2 f 1 mn, (d (x 1 ), d (x 2 ),, d (x n 1 )) p (2) χ 2Rf 1 mn, (d (x 1 ), d (x 2 ),, d (x n 1 )) p χ 2Rf 2 mn, (d (x 1 ), d (x 2 ),, d (x n 1 )) p χ 2Rf 1+f 2 mn, (d (x 1 ), d (x 2 ),, d (x n 1 )) p (3) χ 2Rf 1 mn (q 1 ), (d (x 1 ), d (x 2 ),, d (x n 1 )) p χ 2Rf 1 mn (q 2 ), (d (x 1 ), d (x 2 ),, d (x n 1 )) p χ 2Rf 1 mn (q 1 + q 2 ), (d (x 1 ), d (x 2 ),, d (x n 1 )) p where q 1 and q 2 are two semi norms (4) I 1 is stronger than q 2, then χ 2Rf 1 mn (q 1 ), (d (x 1 ), d (x 2 ),, d (x n 1 )) p χ 2Rf 1 mn (q 2 ), (d (x 1 ), d (x 2 ),, d (x n 1 )) p Proof: We have to take one condition and then easily prove to other condition Hence it is trivial Therefore omit the proof

11 Vector Valued multiple of χ 2 over p metric 97 Competing Interests The authors declare that there is no conflict of interests regarding the publication of this research paper Acknowledgement The research work of the second author Deepmala is supported by the Science and Engineering Research Board (SERB), Government of India under SERB NPDF scheme, File Number: PDF/2015/ The third author wish to thank the Department of Science and Technology, Government of India for the financial sanction towards this work under FIST program SR/FST/MSI-107/2015 References 1 TJI A Bromwich, An introduction to the theory of infinite series, Macmillan and CoLtd, New York, (1965) 2 GH Hardy, On the convergence of certain multiple series, Proc Camb Phil Soc, 19 (1917), F Moricz, Extentions of the spaces c and c 0 from single to double sequences, Acta Math Hung, 57(1-2), (1991), F Moricz and BE Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math Proc Camb Phil Soc, 104, (1988), M Basarir and O Solancan, On some double sequence spaces, J Indian Acad Math, 21(2) (1999), BC Tripathy, On statistically convergent double sequences, Tamkang J Math, 34(3), (2003), A Turkmenoglu, Matrix transformation between some classes of double sequences, J Inst Math Comp Sci Math Ser, 12(1), (1999), A Gökhan and R Çolak, The double sequence spaces c P 2 (p) and c P 2 B (p), Appl Math Comput, 157(2), (2004), A Gökhan and R Çolak, Double sequence spaces l 2, ibid, 160(1), (2005), M Zeltser, Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ of Tartu, Faculty of Mathematics and Computer Science, Tartu, M Mursaleen and OHH Edely, Statistical convergence of double sequences, J Math Anal Appl, 288(1), (2003), B Altay and F Ba aar, Some new spaces of double sequences, J Math Anal Appl, 309(1), (2005), F Başar and Y Sever, The space L p of double sequences, Math J Okayama Univ, 51, (2009), N Subramanian and UK Misra, The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math, 46, (2010) 15 IJ Maddox, Sequence spaces defined by a modulus, Math Proc Cambridge Philos Soc, 100(1) (1986),

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