ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS

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1 It. J. Cotep. Math. Sci., Vol. 1, 2006, o. 1, ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS Qaaruddi ad S. A. Mohiuddie Departet of Matheatics, Aligarh Musli Uiversity Aligarh , Idia Abstract I this paper we characterize the atrix classes (l(p, u),f ) ad (l(p, u),f) which geeralize the atrix classes give by Mursalee [6]. Matheatics Subject Classificatio: 40C05, 40H05 Keywords: Baach liit, alost covergece, atrix trasforatio. 1. Itroductio Let l ad c be the Baach spaces of bouded ad coverget sequeces x =(x ) respectively with the usual or x = sup x. A cotious liear futioal φ o l is called a Baach liit [1] if (i) φ(x) 0 for x =(x ), x 0 for every, (ii) φ(x +1 )=φ(x ), ad (iii) φ(e) = 1 where e =(1, 1, 1,...). A sequece x l is said to be alost coverget to the value L if all of its Baach liits equal to L (see [3]). We deote the set of all alost coverget sequeces by f, i.e. f := {x l : li t (x) =L, uiforly i }, where t (x) = 1 x +,t 1, =0, +1=0 ad L = f li x Nada [7] has defied a ew set of sequeces as follows f := {x l : sup t (x) < },

2 40 Qaaruddi ad S. A. Mohiuddie we call f the set of all alost bouded sequeces. Bullet ad Caar [2] have defied a sequece space l(p, s) ad characterized the atrix classes (l(p, s),l )ad (l(p, s),c). I this paper we chaterize the atrix classes (l(p, s),l ) ad (l(p, s),f), where l(p, u) is ore geeral tha l(p, s). Let p =(p ) be a sequece of real ubers with p > 0. The space l(p, s) is defied as (see [2]) l(p, s) :={x : s x p < } Let u =(u ) be a sequece of real ubers such that u 0( =1, 2,...) ad u 1 =(u 1 ). The space l(p, u) is defied as (see [5]) l(p, u) :={x : u x p < }. If we tae u =(u ) defied by u = s/p, s 0, =1, 2,... the l(p, u) reduces to l(p, s). Obviously x l(p, u) has sae eaig as u.x l(p), ad so l(p, u) is paraored by g(x) =[ u x p ] 1/M, M = ax(1, sup p ). Let X ad Y be two sequece spaces ad A =(a ),=1 be a ifiite atrix of real or coplex ubers. We write Ax =(A (x) ifa (x) = a x coverges for each. Ifx =(x ) X iplies that Ax Y, the we say that A defies a atrix trasforatio fro X ito Y.By (X, Y ) we deote the class of atrices A such that Ax Y for x X. 2. MAIN RESULTS Throghout the text, we use the followig otatio : For all itegers, 1, t (Ax) = 1 +1 A +i (x) i=0 = a(,, )x

3 ALMOST CONVERGENCE AND SOME where a(,, ) = 1 +1 a +i, i=0 Theore 2.1. Let 1 <p < sup p = H < for every.the A (l(p, u),f ) if ad oly if there exists a iteger N>1such that (2.1.1) sup, a(,, ) q u q N q <. Proof. Sufficiecy. Let (2.1.1) hold ad that x l(p, u) usig the followig iequality (see [4]) ab C( a q C q + b p ) for C>0 ad a, b two coplex ubers, we have t (Ax) = a(,, )u 1 u x N[ a(,, ) q u q N q + u x p ] Taig the supreu over, o both sides ad usig (2.1.1), we get Ax f for x l(p, u), i.e. A (l(p, u),f ). Necessity. Let A (l(p, u),f ). Write q (x) = sup t (Ax). It is easy to see that for 0, q is a cotiuous seior o l(p, u) ad (q )is poitwise bouded o l(p, u). Suppose that (2.1.1) is ot true. The there exists x l(p, u) with sup q (x) =. By the priciple of codesatio of sigularities [8], the set {x l(p, u) : sup q (x) = } is of secod category i l(p, u) ad hece oepty, that is, there is x l(p, u) with sup q (x) =. But this cotradicts the fact that (q ) is poitwise bouded o l(p, u). Now by the Baach-Steihauss theore, there is costat M such that (2.1.2) q (x) Mg(x). Now defie a sequece x =(x )by x = { δ M/p (sg a(,, )) a(,, ) q 1 u 1 S 1 N q /p, for > 0 ;

4 42 Qaaruddi ad S. A. Mohiuddie where 0 <δ<1 ad S = 0 =1 a(,, ) q N q. The it is easy to see that x l(p, u) ad g(x) δ. Applyig this sequece to (2.1.2) we get the coditio (2.1.1). Theore 2.2. Let 1 < p sup p = H < for every. The A (l(p, u),f) if ad oly if (i) the coditio (2.1.1) of Theore 2.1 holds, (ii) li a(,, ) =α uiforly i, for every. Proof. Sufficiecy. Let (i) ad (ii) hold ad x l(p, u). For j 1 j a(,, ) q N q u q =1 Therefore sup α q N q u q = li j li j a(,, ) q =1 N q u q < for every. a(,, ) q N q u q sup a(,, ) q N q u q <. Cosequetly the series a(,, )x ad α x coverge for every, ; ad for every x l(u, p). Now, for ɛ>0 ad x l(u, p), choose 0 1 such that 0 +1 By(ii), there exists 0 such that 0 u 1 x p 1 [a(,, ) α ] < =1 for every > 0. By coditio (i), it follows that [a(,, ) α ] 0 +1 is arbitrarily sall.therefore ( ) li a(,, )x = α x uiforly i. Hece A (l(p, u),f).

5 ALMOST CONVERGENCE AND SOME Necessity. Let A (l(p, u),f). Sice f f (see [6]), coditio (i) follows by Theore 2.1. Sice e l(p, u), coditio (ii) follows iediately by (*). Refereces [1] S. Baach, Theòrie des operatios liéaires, Warsaw, [2] E. Bullut, ad O, Çaar, The sequece space l(p, s) ad related atrix trasforatios,co. Fac. Sci. Uiv. Auara (A 1 ), 28 (1979), [3] G. G. Loretz, A cotributio to the theory of diverget sequeces, Acta Math., 80 (1948), [4] I. J. Maddox, Cotiuous ad Kőthe-Toeplitz dual of certai sequece spaces, Proc. Cab. Phil. Soc., 65 (1969), [5] Mursalee ad A. A. Kha, The sequece space l(p, s) ad soe atrix trasforatio, J. Fac. Educatio, 1 (1) (1994), [6] Mursalee, Ifiite atrices ad alost coverget sequeces, Southeast Asia Bull. Math., 19 (1) (1995), [7] S. Nada, O soe sequece spaces, Math. Studet, 48 (4) (1980), [8] K. Yosida, Futioal Aalysis, Spriger-Verlag, Berli Heidelberg, New Yor, Received: Septeber 10, 2005