and ON SOME NEW DOUBLE SEQUENCE SPACES OF INVARIANT MEANS DEFINED BY ORLICZ FUNCTIONS and VAKEEL A. KHAN AND SABIHA TABASSUM

C o m m u n .Fa c .S c i.U n iv .A n k .S e rie s A 1 Vo lu m e 6 0 , N u m b e r 2 , P a g e s 1 1 –1 9 (2 0 1 1 ) IS S N 1 3 0 3 –5 9 9 1 and ON SOME NEW DOUBLE SEQUENCE SPACES OF INVARIANT MEANS DEFINED BY ORLICZ FUNCTIONS and VAKEEL A. KHAN AND SABIHA TABASSUM Abstract. The sequence space BV was introduced and studied by Mursaleen[14]. In this paper we extend BV to 2 BV (p; r; s) and study some properties and inclusion relations on this space. 1. Introduction Let l1 ; and c denote the Banach spaces of bounded and convergent sequences x = (xi ), with complex terms, respectively, normed by kxk1 = sup jxi j, where i i 2 N: Let be an injection of the set of positive integers N into itself having no …nite orbits that is to say, if and only if, for all i = 0; j = 0; j (i) 6= i and T be the 1 operator de…ned on l1 by (T (xi )1 i=1 ) = (x (i) )i=1 : A continuous linear functional -mean if and only if on l1 is said to be an invariant mean or (1) (x) 0; when the sequence x = (xi ) has xi 0 for all i; (2) (e) = 1; where e = f1; 1; 1; :::::::g and (3) (x (i) ) = (x) for all x 2 l1 : If x = (xi ) write T x = (T xi ) = (x (i) ): It can be shown that V = x = (xi ) : 1 X tm;i (x) = L uniformly in i; L = lim x (1) m=1 where m o; i > 0: Received by the editors Agu. 01, 2011, Accepted: Dec. 26, 2011. 2000 Mathematics Subject Classi…cation. 46E30, 46E40, 46B20. Key words and phrases. Invariant means, double sequence spaces, Orlicz Function. c 2 0 1 1 A n ka ra U n ive rsity 11 12 VAKEEL A. KHAN AND SABIHA TABASSUM 1 tm;i (x) = xi + x (i) + :::: + x m (i) m+1 and t 1;i = 0 (2) m . Where (i) denote the mth iterate of (i) at i: In the case is the translation mapping, (i) = i + 1 is often called a Banach limit and V ; the set of bounded sequences of all whose invariant means are equal, is the set of almost convergent sequence. Subsequently invariant means have been studied by Ahmad and Mursaleen[1], Mursaleen[12,13], Raimi[15] and many others. The concept of paranorm is closely related to linear metric spaces. It is generalization of that of absolute value. Let X be a linear space. A Paranorm is a function g : X ! R which satis…es the following axioms: for any x; y; x0 2 X, ; 0 2 C, (i) g( ) = 0; (ii) g(x) = g( x); (iii) g(x + y) g(x) + g(y) (iv) the scalar multiplication is continuous, that is ! 0 , x ! x0 imply x ! 0 x0 : Any function g which satis…es all the condition (i)-(iv) together with the condition (v) g(x) = 0 if only if x = , is called a Total Paranorm on X and the pair (X; g) is called Total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (cf.[18],Theorm 10.42,p183]) An Orlicz Function is a function M : [0; 1) ! [0; 1) which is continuous, nondecreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) ! 1, as x ! 1. If convexity of M is replaced by M (x + y) M (x) + M (y) then it is called Modulus function. An Orlicz function M satis…es the 2 exist constant k 2 and u0 > 0 such that M (2u) whenever juj condition (M 2 2 for short ) if there KM (u) u0 . 1 The second author is supported by Maulana Azad National Fellowship under the University Grants Commision of India. ON SOM E NEW DOUBLE SEQUENCE SPACES OF INVARIANT M EANS 13 An Orlicz function M can always be represented in the integral form Rx M (x) = q(t)dt; where q known as the kernel of M; is right di¤erentiable for 0 t 0; q(t) > 0 for t > 0; qis non-decreasing and q(t) ! 1 as t ! 1: Note that an Orlicz function satis…es the inequality M ( x) M (x) for all with 0 < < 1; since M is convex and M (0) = 0: W.Orlicz used the idea of Orlicz function to construct the space (LM ): Lindesstrauss and Tzafriri [9] used the idea of Orlicz sequence space; lM := x2w: 1 X M jxk j k=1 < 1; for some which is Banach space with the norm the norm 1 X jxk j kxkM = inf >0: M >0 1 : k=1 The space lM is closely related to the space lp , which is an Orlicz sequence space with M (x) = xp for 1 p < 1: Orlicz functons have been studied by V.A.Khan[3,5,6,7,8] and many others. Throughout a double sequence is denoted by x = (xij ):A double sequence is a double in…nite array of elements xij 2 R for all i; j 2 N: Let 2 l1 and 2 c denote the Banach spaces of bounded and convergent double sequence x = (xi;j ) respectively. Doube sequence spaces have been studied by Moricz and Rhoads[11], E.Savas and R.F.Patterson[16], V.A.Khan[4] and many others. Let l by 2 1 be an injection having no …nite orbits and T be the operator de…ned on 1 T ((xi;j )1 i;j=1 ) = (x (i;j) )i;j The idea of -convergence for double sequences has recently been introduced in [2] and further studied by Mursaleen and Mohiuddine [12]. For double sequences, 1 X 1 X V = x = (x ) : tmnpq (x) = L uniformly in p; q; L = lim x see[16] 2 i;j m=1 n=1 (3) tmnpq (x) = 1 X 1 X 1 x i (p); j (q) ; p; q = 0; 1; 2::: (m + 1)(n + 1) i=1 j=1 (4) 14 VAKEEL A. KHAN AND SABIHA TABASSUM t0;0;p;q (x) = xpq ; t 1;0;p;q (x) = xp 1;q (x); t0; 1;p;q (x) = xp;q 1 ; t 1; 1;p;q (x) = xp 1;q 1 ; and x i (p); j (q) = 0 for all i or j or both negative. A double sequence space E is said to be solid if ( i;j xi;j ) 2 E, whenever (xi;j ) 2 E, for all double sequences ( i;j ) of scalars with j i;j j 1; for all i; j 2 N: Let K = f(ni ; kj ) : i; j 2 N; n1 < n2 < n3 < :::: and k1 < k2 < k3 < :::g N N and E be a double sequence space. A K-step space of E is a sequence space E K = f( i;j xi;j ) : (xi;j ) 2 Eg: A canonical pre-image of a sequence (xni ;kj ) 2 E is a sequence (bn;k ) 2 E de…ned as follows: ( ank if (n; k) 2 K; bnk = 0 otherwise : A canonical pre-image of step space elements in E K: E K is a set of canonical pre-images of all A double sequence space E is said to be monotone if it contains the canonical pre-images of all its step spaces. A double sequence space E is said to be symmetric if (xi;j ) 2 E implies (x (i); (j) ) 2 E; where is a permutation of N: 2. Main Results Lemma 1 A sequence space E is solid implies E is monotone. Mursaleen[14] de…ned the sequence space BV = fx 2 l1 : where X m j m;i (x)j < 1; uniformly in ig; m;i (x) = tm;i (x) assuming that tm;i (x) = 0 for m = 1 A straightforward calculation shows that tm 1;i (x) (5) ON SOM E NEW DOUBLE SEQUENCE SPACES OF INVARIANT M EANS m;n (x) = We de…ne 2 BV 8 < : 1 m(m+1) n[xn (i) xn 1 (i)] (m 1) (6) n=1 xi (m = 0): = fx 2 2 l1 : where mnpq (x) = m P 15 X m;n j mnpq (x)j < 1; uniformly in p and qg; 8 m P n P > 1 > ij[x i (p); j (q) > m(m+1)n(n+1) < i=1 j=1 (7) x i 1 (p); j (q) x i (p); j 1 (q) + x i 1 (p); j 1 (q) ] (m; n > > > :x m or n or both zero : (see[12]) 1) (8) ij Let M be an Orlicz function, p = (pi ) be any sequence of strictly positive real numbers and r 0. V.A.Khan[5] de…ned the following sequence space: BV (M; p; r) = x = (xi ) : 1 X j m;i (x)j 1 M r m m=1 uniformly in i and for some pi < 1; >0 : Let p = (pij ) be any double sequence of strictly positive real numbers a (...truncated)