Faulty of Siees ad Matheatis, Uivesity of Niš, Sebia Available at: http://www.pf.i.a.yu/filoat Filoat 22:2 (28), 59 64 SOME NEW SEQUENCE SPACES AND ALMOST CONVERGENCE Saee Ahad Gupai Abstat. The sequee spae a have bee defied ad the lasses ( a : l p) ad ( a : ) of ifiite aties have bee haateized by Aydi ad Başa (O the ew sequee spaes whih ilude the spaes ad, Hoaido Math. J. 33(2) (24), 383-398) [], whee p. The : f a :, whee ai pupose of the peset pape is to haateize the lasses ( a ) ad ( f ) f ad f deote the spaes of alost oveget ad alost oveget ull sequees with eal o oplex tes. 2 Matheatis Subjet Classifiatio: 46B2; 46E3; 4C5 Keywods ad phases. Sequee spaes of o absolute type, alost oveget sequees, β duals ad atix appigs. Reeived : July 2, 28 Couiated by Vladii Raočević. Itodutio Let ω be the spae of all sequees, eal o oplex ad let l ad espetively be the Baah spaes of bouded ad oveget sequees x = ( x ) with the usual o x = sup x S : l l ( Sx) x +. Let be the shift opeato defied by = fo all. A Baah liit L is defied o l, as a o egative liea futioal suh that L ( Sx ) = L( x ) ad Le () =, e = (,,,... ) [2]. A sequee x l is said to be alost oveget to the geealized liit α if all Baah liits of x ae α [3]. We deote the set of alost oveget sequees by f ad alost oveget ull sequees by f, i.e. f = x l :li t ( x) = α, uifoly i { } ad f = { x l :li t( x) =, uifoly i } Whee ad t( x) = x +, + = t, = α = f li x.
6 Let λ ad μ be two sequee spaes ad A= ( a ) be a ifiite atix of eal o oplex ubes a, whee, = {,, 2,... }. The we say that A defies a atix appig fo λ i to μ, ad deote it by witig A : λ = ( ), the sequee Ax = ( Ax) ( Ax) = a x, ( ) (.) x x λ { } μ if fo evey sequee, the A-tasfo of x, is i μ, whee Fo sipliity i otatio, hee ad i what follows, the suatio without liits us fo to. We deote by ( λ : μ ) the lass of all aties A suh that A : λ μ. Thus A ( λ : μ ) if ad oly if the seies o the ight side of (.) oveges fo evey ad evey x λ. Fo a sequee spae λ, the atix doai λ A of a ifiite atix A is defied by λ = x = x ω : Ax λ A { ( ) } a f of ifiite aties. The sequee spae a is defied as the set of all sequees whose A - tasfo is i [], i.e. a = x = ( x) ω :li ( ) x exists + + = Whee A deotes the atix A = ( a ) defied by +, ( ) a = +, ( > ) We efe the eade to [] fo elevat teiology ad additioal efeees o the spae. a sequee The objet of this pape is to haateize the lasses ( a : f ) ad ( : ) 2. Mai Results Defie the sequee y = ( y () ), whih will be used as the A - tasfo of a x = ( x ), i.e. j + y () = x ; + j ( ) j = Fo bevity i otatio, we wite (2.)
6 a (,, ) = a+ j, + j = ad a (,, ) a (,, ) a (, +, ) a %(,, ) = Δ + = + + + + + fo,,. We deote by ( ) ( ) β λ, the β -dual of a sequee spae λ ad ea the set of the x x y = y λ. Now, we ay give the followig lea whih is eeded i povig the Theoe (2.) below. sequees = ( ) suh that x y = ( x y ) s fo all ( ) Lea 2.[]: Defie the sets d ad d2 as follows a d = a = ( a ) ω : Δ ( + ) < + a d2 = a = a ω : s + a a a + whee Δ = fo all. + + + + ad ( ) β a = d d 2 The. ) Theoe 2.: A ( a : f if ad if sup a % (,, ) < (2.2), a s fo all. (2.3) + li a %(,, ) = α uifoly i, fo eah (2.4) li a (,, ) α = % uifoly i. (2.5) Poof: Suppose that the oditios (2.2), (2.3), (2.4) ad (2.5) hold ad exists ad at this stage, we obseve fo (2.4) ad (2.2) that α j sup a % (, j, ) < j=, j x a. The Ax
62 holds fo evey a, we have y. This gives that ( α ) l. Sie x a by the hypothesis, ad α y. Theefoe, oe a easily see that ( ) y ad also thee exists M > suh that sup y hoose a fixed, thee is soe % = [ a (,, ) α ] fo evey, uifoly i. Also, by (2.5), thee is soe fo evey = + y < a % (,, ) α < by (2.4) suh that ε 2, suh that ε 2 M uifoly i. Theefoe, we have ( ) α = [ (,, ) α ] + i + % Ax y a y i= l fo eah < M. Now fo ay ε >, [ %(,, ) α ] [%(,, ) α ] a y + a y = = + ε < + a (,, ) α 2 % = + ε ε < + M = ε 2 2 M fo all suffiietly lage, uifoly i. Hee Ax f, whih poves the suffiiey. this iplies th Covesely suppose that A ( a : f ) at { } y. The Ax exists fo evey x a ad a a β fo eah ; the eessity of (2.3) is iediate. Now a(,, ) x exists fo eah, ad { } x a, the sequee a = a(,, ) defie the otiuous liea futioals φo a by φ ( x) = ax (,, ), (, ). Sie a ad ae o isoophi ([], Theoe 2.2), it should follow with (2.) that φ = % a
63 This just says that the futioals defied by φ o ae poit wise bouded. Hee, by the Baah- Steihauss theoe, they ae uifoly bouded, whih yields that thee exists a ostat M > suh that φ M fo all, It theefoe follows, usig the oplete idetifiatio just efeed to that a % (,, ) = φ M, holds fo all whih shows the eessity of the oditio (2.2). To pove the eessity of (2.4), oside the sequee ( ) ( b = b ) a fo evey, whee { } ( ) ( ) + ( ) ( ), ( ) + b ( ) = + ;,, ( o > + ) Sie Ax exists ad is i f fo eah x a, oe a easily see that ( ) a Ab () = Δ ( ) f + + fo eah, whih shows the eessity of (2.4). Siilaly by taig x = e a, we also obtai that a Ax = Δ ( ) f + + ad this shows the eessity of (2.5). This opletes the poof. If the spae f is eplaed by f, the Theoe (2.) is edued to Coollay 2.: A ( a : f if ad if (2.2), (2.3) ad (2.4), (2.5) also hold with α = fo all. ) Refeees [] C. Aydi, Feyzi Başa, O the ew sequee spaes whih ilude the spaes ad, Hoaido Math. J. 33 (2)(24), 383-398. [2] S. Baah, Theòie des opeatios liéaies, Waszawa, 932. [3] G. G. Loetz, A otibutio to the theoy of diveget sequees, Ata Math. a ( )
64 8(948), 67-9. [4] S. Nada, Matix tasfoatios ad alost boudedess, Glasi Mat. (34), 4(979), 99-7. Depatet of Matheatis, Natioal Istitute of Tehology, Siaga-96, Idia E-ail: saeegupai@ediffail.o