Matrix transformations between the spaces of Cesàro sequences and invariant means

CESÀRO SEQUENCES AND INVARIANT MEANS

Received 6 March 2005; Revised 16 October 2005; Accepted 18 December 2005

The main purpose of this paper is to characterize the classes of matrices (ces[(p),(q)],cσ) and (ces[(p), (q)],l_∞σ), wherecσ is the space of all bounded sequences all of whose σ -means are equal,l_∞σ is the space ofσ- bounded sequences, and ces[(p), (q)] is the gener-alized Ces`aro sequence space.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction

Letω be the space of all sequences, real or complex, and letl∞ and c, respectively, be the Banach spaces of bounded and convergent sequences x=(xn) with norm x =

sup_k≥0|xk|. Let σ be a mapping of the set of positive integers into itself. A

continu-ous linear functionalφonl∞ is said to be an invariant mean or aσ- mean if and only if (i)φ(x)≥0, when the sequencex=(xn) hasxn≥0 for each n; (ii)φ(e)=1, where

e=(1, 1, 1,...); and (iii)φ((xσ(n)))=φ(x),x∈l∞.

For certain kinds of mappings, everyσ- mean extends the limit functionalφoncin the sense thatφ(x)=limxforx∈c(see [2,15]). Consequently,c⊂cσ_{, wherec}σ _{is the}

set of bounded sequences, all of whose invariant means are equal (see [1,9,10]). Whenσ is translation, theσ- means are classical Banach limits onl∞(see [2]) andcσ is the set of almost convergent sequencesc(see [7]). Almost convergence for double sequences was introduced and studied by M ´oricz and Rhoades [8] and further by Mursaleen and Savas¸ [13], Mursaleen and Edely [12], and Mursaleen [11].

Ifx=(xn), writeTx=(Txn)=(xσ(n)), then

cσ_=x∈l

∞: lim_m→∞tm,n(x)=L, uniformly inn,L=σ−limx

, (1.1)

where

tm,n(x)=_m1

+ 1

m

j=0

Tj_x

n withTjxn=xσj_(n),t_−1,n(x)=0. (1.2)

We definelσ_∞the space ofσ- bounded sequences (Ahmad et al. [2]) in the following way.

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 74319, Pages1–8

Letxn=z0+z1+z2+···+znand

lσ

∞=

z∈ω: sup

m,n

_{ψm,n(z)<∞, (1.3)}

where

ψm,n(z)=tm,n(x)−tm−1,n(x)

=_m 1

(m+ 1)

m

j=1

j

hj

i=hj−1+1

zi, hj=σj(n).

(1.4)

Ifσ(n)=(n+ 1), thenl_∞σ is the set of almost bounded sequencesl∞(see [14]). LetA=(ank) be an infinite matrix of complex numbersank(n,k=1, 2,...) andX,Y two subsets ofω. We say that the matrixAdefines a matrix transformation fromXinto Y if for every sequencex=(xk)∈X the sequenceA(x)=(An(x))∈Y, whereAn(x)=

kankxk converges for eachn. We denote the class of matrix transformations from X intoY by (X,Y).

The main purpose of this paper is to characterize the classes (ces[(p), (q)],cσ) and (ces[(p), (q)],lσ_∞) and deduce some known and unknown interesting results as corollaries. The classes (ces[(p), (q)],cσ) and (ces[(p), (q)],lσ_∞) are due to Khan and Rahman [4]. If{qn}is a sequence of positive real numbers, then for p=(pr) with infpr>0, we

define the space ces[(p), (q)] by

ces (p), (q)=

x∈ω:

∞

r=0

1 Q2r

r qk

_xkpr

<∞

, (1.5)

whereQ2r=q2r+q2r+1+···+q2r+1₋₁and_rdenotes a sum over the range 2r≤k <2r+1. Remark 1.1. Ifqn=1 for alln, then ces[(p), (q)] reduces to ces(p) studied by Lim [6].

Also, ifpn=pfor allnandqn=1 for alln, then ces[(p), (q)] reduces to cespstudied by

Lim [5].

For any bounded sequence p, the space ces[(p), (q)] is a paranormed space with the paranorm given by (see [4])

g(x)=

_∞

r=0

1 Q2r

r qk

_xkpr1/M

(1.6)

2. Sequence-to-sequence transformations

In this section, we characterize the classes (ces[(p), (q)],cσ) and (ces[(p), (q)],lσ_∞). We writea(n,k) to denote the elementsankof the matrixA, and for all integersn,m≥

1, we write

tmn(Ax)=Axn+TAxn+···+T m_Axn

m+ 1

=

k

t(n,k,m)xk, (2.1)

wheret(n,k,m)=1/(m+ 1)mj=0a(σj(n),k).

We also define the spaces ofσ- convergent series andσ- bounded series, respectively, as follows: cσ s = x: m

i=1

1 i+ 1

i

j=0

xσj_(n)

is convergent uniformly inn, asm−→ ∞

,

bσ s =

x: sup

n,m m

i=1

1 i+ 1

i

j=0

xσj_(n)

<∞

.

(2.2)

If we takeσ(n)=n+ 1,cσs andbσs reduce tocsandbs, as defined below:

cs=

x:

m

i=1

1 i+ 1

i

j=0

xj+n

is convergent uniformly inn, asm−→ ∞

,

bs=

x: sup

n,m m

i=1

1 i+ 1

i

j=0

xj+n

<∞

.

(2.3)

Now we prove the following theorem.

Theorem 2.1. Let 1< pr≤suprpr<∞. ThenA∈(ces[(p), (q)],cσ) if and only if

(i) there exists an integerE >1 such that for alln,

U(E)=sup

m

∞

r=0

Q2rmax

r

_t(n,k,m) qk

tr

E−tr<_{∞, (2.4)}

where 1/pr+ 1/tr=1,r=0, 1, 2,..., and maxrmeans maximum over 2r≤k <2r+1;

(ii)a(k)=(ank)∞n=1∈cσfor eachk, that is, limmt(n,k,m)=ukuniformly inn, for each

k.

In this case,σ-limit ofAxis∞_k=1ukxk.

Proof

Neccessity. Suppose thatA∈(ces[(p), (q)],cσ). Now∞_k=1t(n,k,m)xk exists for eachm

Therefore, it follows that each{fm,n}mdefined by

fm,n(x)=tm,n(Ax) (2.5)

is an element of ces∗[(p), (q)]. Since ces[(p), (q)] is complete and further for eachn, sup_m|tm,n(Ax)|<∞on ces[(p), (q)]. Now arguing with the uniform boundedness

prin-ciple, we have condition (i). Sinceek∈ces[(p), (q)], condition (ii) follows.

Sufficiency. Suppose that the conditions hold. Fixn∈N. For every integers≥1, from (i) we have

s

r=0

Q2rmax_r

q−1

k t(n,k,m) tr

E−tr_≤sup

m

∞

r=0

Q2rmax_r (q−k1t(n,k,m) tr

E−tr.

(2.6)

Now lettings→ ∞, we obtain

limm→∞ ∞

r=0

Q2rmax

r (q− 1

k t(n,k,m) tr

E−tr_≤sup

m

∞

r=0

Q2rmax

r

q−1

k t(n,k,m) tr

E−tr.

(2.7)

Therefore, from (ii) we have

∞

r=0

Q2rmax

r

q−1

k uk

tr

E−tr_≤sup

m

∞

r=0

Q2rmax

r

q−1

k t(n,k,m))

tr

E−tr<_∞. _(2.8)

Hence (uk)kand{t(n,k,m)}k∈ces∗[(p), (q)], therefore the series∞k=1t(n,k,m)xk and

_∞

k=1ukxk converge for each m and n and x∈ces[(p), (q)]. For given>0 and x∈

ces[(p), (q)], choosessuch that

_∞

r=s+1

1 Q2r

r qk

_xkpr1/M

<. (2.9)

Since (ii) holds, there existsm0such that

s

k=1

t(n,k,m)−uk

< ∀m > m0. (2.10)

Since (i) holds, it follows that

∞

k=s+1

t(n,k,m)−uk

Therefore,

lim_m ∞

k=1

t(n,k,m)xk=

∞

k=1

ukxk, uniformly inn. (2.12)

This completes the proof.

Remark 2.1. For different choices of p,q, andσ, we can deduce many corollaries from the above theorem to characterize the matrix classes, for example, (ces(p),cσ), (cesp,cσ),

(cesp(q),cσ), (ces[(p), (q)],c), and so forth. The class (ces(p),c) was characterized by F.

M. Khan and M. A. Khan [3] which we can obtain directly from our theorem by taking qn=1 for allnandσ(n)=n+ 1.

We write (see [2])

x0=z0+z1+···+zn,

ψm,n(Az)=

k

α(n,k,m)zk, (2.13)

where

α(n,k,m)=_m 1

(m+ 1)

m

j=1

j hj

i=hj−1+1 aik

, hj=σj(n). (2.14)

Now we prove the following theorem.

Theorem 2.2. Let 1< pr≤suprpr<∞. ThenA∈(ces[(p), (q)],lσ∞) if and only if

sup

m,n

∞

r=0

Q2rmax

r

q−1

k α(n,k,m) tr

E−tr<_{∞, (2.15)}

whereEis an integer greater than 1 and 1/pr+ 1/tr=1,r=0, 1, 2,....

Proof

Necessity. Suppose thatA∈(ces[(p), (q)],lσ_∞). Nowk∞=1α(n,k,m)zk exists for eachm

andnandz∈ces[(p), (q)], whence{α(n,k,m)}k∈ces∗[(p), (q)] for eachmandn. There-fore, it follows that{fm,n}defined by

fm,n(x)=ψm,n(Az) (2.16)

is an element of ces∗[(p), (q)]. Since ces[(p), (q)] is complete and further sup_m,n|ψm,n(Az)|

Sufficiency. Suppose that condition (2.15) holds. Fixn∈N. For every integers≥1 we have

s

r=0

Q2rmax_r

q−1

k α(n,k,m) tr

E−tr_≤sup

m,n

∞

r=0

Q2rmax_r

q−1

k α(n,k,m) tr

E−tr.

(2.17)

So

lim

s→∞

s

r=0

Q2rmax_r

q−1

k α(n,k,m) tr

E−tr

≤sup

m,n

∞

r=0

Q2rmax_r

q−1

k α(n,k,m) tr

E−tr<_∞.

(2.18)

Hence{α(n,k,m)} ∈ces∗[(p), (q)]. Therefore, the series∞k=1α(n,k,m)zkconverges for

eachmandnandz∈ces[(p), (q)].

This completes the proof.

Remark 2.2. The matrix class (ces(p),l∞), was characterized by F. M. Khan and M. A. Khan [3] which we can obtain directly from the above theorem by lettingqn=1 for alln andσ(n)=n+ 1. Besides, we can further deduce many corollaries for different choices of p,q, andσ.

3. Sequence-to-series transformations

For all integersm,n≥1, we write

t∗

mn(Ax)= m

i=1

tin(Ax)=

k m

i=1

1 i+ 1

i

j=0

aσj_(n),kx

k=

k

t∗_{(m,n,k)xk, (3.1)}

where

t∗_(m,n,k)=m

i=1

1 i+ 1

i

j=0

aσj_{(n),k. (3.2)}

Theorem 3.1. Let 1< pr≤suprpr<∞. ThenA∈(ces[(p), (q)],cσs) if and only if

(i) there exists an integerE >1 such that for alln,

U(E)=sup

m

∞

r=0

Q2rmax_r t

∗_(n,k,m) qk

tr

E−tr<_{∞, (3.3)}

(ii)a(k)= {ank}∞n=1∈cσs for eachk, that is, limmt∗(n,k,m)=ukuniformly inn, for each

k.

In this case, theσ-limit ofAxis∞k=1ukxk.

Theorem 3.2. Let 1< pr≤suprpr<∞. ThenA∈(ces[(p), (q)],bσ_∞) if and only if

sup

m,n

∞

r=0

Q2rmax

r

q−1

k t∗(n,k,m) tr

E−tr<_{∞, (3.4)}

whereEis an integer greater than 1 and 1/pr+ 1/tr=1,r=0, 1, 2,....

Proofs of Theorems3.1and3.2are similar to those of Theorems2.1and2.2, respec-tively.

Remark 3.1. Ifσis translation, then Theorems3.1and3.2give the characterization for the classes (ces[(p), (q)],cs) and (ces[(p), (q)],bs). As Remarks2.1and2.2, for different choices ofp,q, andσ, we can deduce many corollaries.

References

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M. Mursaleen: Department of Mathematics, Faculty of Science, King Abdul-Aziz University, P.O. Box 80203, Jeddah, The Kingdom of Saudi Arabia

E-mail address:[email protected]

E. Savas¸: Department of Mathematics, Y. Y. University, 65080 Van, Turkey

E-mail address:[email protected]

M. Aiyub: Department of Mathematics, Aligarh Muslim University (AMU), Aligarh 202002, India

E-mail address:mohdaiyub@rediffmail.com

S. A. Mohiuddine: Department of Mathematics, Aligarh Muslim University (AMU), Aligarh 202002, India