Infinite Matrices and Almost Bounded Sequences

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Vietnam J Math (2014) 42:153-157 DOI 10 1007/sl 0013-013-00«M

Infinite Matrices and Almost Bounded Sequences

A. Hamid Ganie • Neyaz Ahmad Sheikh

Received 14 March 2013 / Accepted. 27 August 2013 / Published online- 8 October 2013

Abstract In the present paper, we charactenze (bv(p): / ^ ) , (bv(p): / ) , and (bv{p): /Q), where /oo, / , and /o denote, respectively, the spaces of almost bounded, almost convergent and almost null sequences and the space bv(p) has recently been studied by Jarrah and Malkowsky (Collect. Math., 55:151-162, 2004).

Keywords Sequence space of non-absolute type • Paranormed sequence space • Almost convergence • ^-Duals and matrix transformations

Mathematics Subject Classification (2000) 46A45 40C05

1 Preliminaries, Background, and Notation

Let w denote the space of all sequences (real or complex). Any subspace of cu is called a sequence space. So a sequence space is the set of scalar sequences (real of complex) which IS closed under coordinate wise addition and scalar multiplication. Throughout the paper N, E, and C denote the set of non-negative integers, the set of real numbers and die sei of complex numbers, respecfively.

Let X, Y be two sequence spaces and let A = {a„^) be an infinite matrix of real or complex numbers «„(, where n,k e N, Then the matrix A defines the A-transformation from X into Y if for every sequence .v = (xt) e X, ihe sequence Ax — ((A.v)n|, the A-transform of X. exists and is in Y, where (Ax)„ = X!i "ntxi. • For simplicity in notation, here and in what follows, the summation without limits runs from 0 to oo By A e (X : Y) we mean the set of matrices from X to Y, i.e., A:X -*• Y. A sequence x is said to be A-summable to / if Ax converges to / which is called the A-lirml of J:.

A.H Ganie|IS)-NA Sheikh

Depanmeni of Mathematics, National Institute of Technology, Srinagar 190006. India e-mail: [email protected]

NA Sheikh e-muil,neyaitnil®yahoo.co.in

fi Springe

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1S4 A.H. Ganie, N,A. Sheikh For a sequence space X. the matrix domain XA of an infinite matrix A is defined as

XA^{X = (X,):AX€X}. (1) Let lac and c be Banach spaces of bounded and convergent (respectively) sequences JC =

{x„]%Q with supremum norm ||J:|| — sup^ ]jrn|. Let T denote the shift operator on to, that is, Tx — [x„)'^^, T-x — lx„]^2 ^^d s° oi^- ^ Banach limit L is defined on /oo as a non- negative Imear functional such that L is shift invariant, i.e., L(Tx) = L(x) and L(e) = 1, e - ( L L . . . ) [ l ) .

Lorentz(see [6]) called a sequence {;Cn| almost convergent if all Banach limits of J:,Z,(X), are same and this unique Banach limit is called f-limit of J: , In his paper, Lorentz proved the following criterion for almost convergent sequences.

A sequence x — [x^} e l^ is almost convergent with F-limit L(x) if and only if lim t„„(x) = L(x),

where t„„(x) ^ ^ ^ ^ J ^ ' T^x,. (T° = 0) uniformly in « > 0, We denote the set of almost convergent sequences by / , Nanda (see [2]) has defined a new set of sequences /cc as follows:

f^=^\x&l^-s\i^\l„,„{x)\ < o o | . We call /co the set of all almost bounded sequences.

An infinite matrix A = (a^,,) is said to be regular or a Toeplitz matrix (see [14]) if and only if the following conditions hold:

(i) l i m „ ^ ^ ^ ^ 0 (i„( = 1, (ii) hm„^;,^ o^t ^ 0 (fc = 0, 1....), (i") E ^ Q \a.k I < A/ (M > 0, y ^ 0, 1,...).

The approach of construcung a new sequence space by means of matrix domain of a particular limitation method has been studied by several authors (see [1-5, 8-13, 15]), Jar- rah and Malkowsky [5] has defined the sequence space bv(p) and characterize the classes ibv(p) : l^), (bvip): c), and {bv(p); CQ). In the present paper, we characterize the classes ibv{p):f^),{bv(p) / ) , a n d ( M p ) /o). The space M p ) is defined [5] as

bv(p)=.\x = {xk)eco:Y^\Axk\'"' < oo , where Ax*. =Xk— Xk-i.

We denote by X^ the ^-dual of a sequence space X and mean the set of all the sequences X ^ (jTt) such that xy ^ (x^y^) € cs for all y = (>•*) e X, where by cs we denote the sequence of all convergent series.

2 Main Results

For brevity in notation, we shall wnle 1

~ m + 1 '_

where o(«./:.m) = ^X:7=o«"+y,*, (n.k,i fl Springer

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Infinite Matrices and Almost Boimded Sequences

Theorem 1 Let I < pk < H < oo for every it e N. Then A e {bv(p): f^) if and only if there exists an integer C > 1 suck that

sup V |a(n,fc,m)p*C-''* < co. (2)

Proof Sufficiency. Suppose the condition (2) holds and x e bv{p}. Then (an.tligpj E [bv(p)]^ for every « e N, the A-transform of x exists. Using the inequality which holds for any C > 0 and any complex numbers a, b that \ab\ < C{\aC\'^ + \b\''], where p>l and p " ' -I- 9 " ' — 1 [7], we have

\l,„J.Ax)\ = \j^a{n,k,m)x^<Y,C{\a{n,k,m}C-'\'' + \x^r],

' k ' k

where p ^ ' T-^^' = 1 Taking supremum over m,« on both sides, we get Ax e fee- Necessity. Suppose A e (bv{p) : faa). Then Ax exists for every x e bv{p) and this implies that [ii„,iligH e [fov(/7)]^ for every K e N. Now, '^^a{n,k,m)xk exists for each m,n e N and x e fcv(p), the sequences {a(n, k,m))k^^ define the continuous linear func- fionals 0mnix) on bv{p) by *mn(jc) = E t ( " ' *^' '")^* ("' "^ ^ ^ ) - ^mce fcv(p) is complete and sup„ „ | ^ ^ a(n, k, m)xk | < co on bv(p), so, by the uniform boundedness prmciple, there exists M > 0 such that

supl0m„(A:)| — sup y^(1 (/?,/:,m)j:j: <M<co

This implies that sup^, „ \'Y^^a{n,k,m)\''i<C'^>< < oo, which shows the necessity of the con-

dition (2) and the proof is complete • Theorem 2 Ut \. < pk < H < oo for every keN. Then A e (bv(p): / ) if and only if (2)

holds and there is a sequence ofscalars (fik) such that

lim a(n,k,m) = fik uniformlyinn^H. (3) Proof Sufficiency. Suppose diat the conditions (2) and (3) hold and x e bv{p). We observe

for y > 1

^\a(n,k,m)\'"C-"' < ^ |a(«,/:, mjI'^^C""'- < oo foreveryn.

j t = 0 »

Therefore

y'lAI''*C"^* sY^\a{n,k,m)\'''-C~'" < oo unifonnly in/i.

Consequently, reasoning as in the proof of the sufficiency of Theorem 1, the series Y.ya{n,k,m)Xi and ^ , ^t converge for every n, m and every JT € M p ) - N o w , for a given e > 0, choose a fixed A:o e N such that

fl Spnnget

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^5g A.H. Ganie, N,A Sheith Then by (3) there is some WIQ e N such that

I * I**

\Y,[a(n,k,m)-M<^

u=o I

for every m > mo uniformly m n. Moreover, smee '^i,a(n,k,m)xt and J^k^k converge (absolutely) uniformly m n. mforxe bv(p), we observe that

Y^[a(n,k,m)-^i,]xk

converges uniformly in «,m and for J: ebv(p). Hence by (2) and (3) we have

\Z[a(n,k,m)-^A<l

\k=0 I for all m > mo and uniformly in n. Therefore,

y^[(i(H,ft:.m) - ^ t ] -*• 0 uniformly in «,

ie.,

limy^a(n,k.m)xk = fik umformlyinM. (4)

k

Hence Ax e / , this proves sufficiency.

Necessity. Suppose that A e (bv(p) : / ) . Then, since f C foo C l^, the necessity of (2) IS immediately obtained from Theorem 1 above. Further, consider the sequence et = (0.0 1.0....) eftv(p). Since Ae* = (a„i,) € f for each fixed A: e N we get the neces-

sity of (3). This completes the prooT D Theorem 3 A € (bv(p) : /o) if and only if (2) and (3) hold along with ^t = 0 for each

ken.

Proof The proof follows from Theorem 2 above by taking ^f ^ - 0 for each t e N. D Acknowledgements We would like to express our gratitude to the referee(s) for the careful reading, valu- able suggestions that improved ihe presentation of the paper.

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Infinite Malnces and Almost Bounded Sequences

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Infinite Matrices and Almost Bounded Sequences