Bounds for the norm of lower triangular matrices on the Cesàro weighted sequence space

R E S E A R C H

Open Access

Bounds for the norm of lower triangular

matrices on the Cesàro weighted sequence

space

Davoud Foroutannia

_{and Hadi Roopaei}

*_{Correspondence:}

[email protected] Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

Abstract

This paper is concerned with the problem of finding bounds for the norm of lower triangular matrix operators fromlp(w) intocp(w), wherecp(w) is the Cesàro weighted sequence space and (wn) is a non-negative sequence. Also this problem is considered for lower triangular matrix operators fromcp(w) intolp(w), and the norms of certain matrix operators such as Cesàro, Nörlund and weighted mean are computed.

Keywords: norm; lower triangular matrix; Nörlund matrix; weighted mean matrix; weighted sequence space

1 Introduction

Letp≥ andωdenote the set of all real-valued sequences. The spacelp is the set of all real sequencesx= (xn)∈ωsuch that

xp= _∞

n=

|xn|p /p

<∞.

Ifw= (wn)∈ωis a non-negative sequence, we define the weighted sequence spacelp(w) as follows:

lp(w) :=

x= (xn)∈ω:

∞

n=

wn|xn|p<∞

,

with norm · p,w, which is defined in the following way:

xp,w= _∞

n= wn|xn|p

/p .

LetC= (cn,k) denote the Cesàro matrix. We recall that the elementscn,kof the matrixC are given by

cn,k= ⎧ ⎨ ⎩ 

n for ≤k≤n,  fork>n.

The sequence space defined by

cp(w) =

(xn)∈ω:Cx∈lp(w)

=

(xn)∈ω:

∞

n= wn

 n

n

i= xi

p

<∞

is called the Cesàro weighted sequence space, and the norm · p,w,cof the space is defined by

xp,w,c= _∞

n= wn

 n

n

i= xi

p/p .

The Cesàro sequence spaces were studied in [], wherewn=  for alln. It is significant that in the special casewn= , we havelp(w) =lpandcp(w) =cp.

Let (wn) be a non-negative sequence andA= (an,k) be a lower triangular matrix with non-negative entries. In this paper, we shall consider the inequality of the form

Axp,w,c≤Uxp,w,

and the inequality of the form

Axp,w≤Uxp,w,c,

wherex= (xn) is a non-negative sequence. The constantUdoes not depend onx, and we seek the smallest possible value ofU. We writeAp,w,cfor the norm ofAas an operator fromlp(w) intocp(w),Ap,cfor the norm ofAas an operator fromlpintocp,Ac,p,wfor the norm ofAas an operator fromcp(w) intolp(w),Ac,pfor the norm ofAas an operator fromcpintolp,Ap,wfor the norm ofAas an operator fromlp(w) into itself andApfor the norm ofAas an operator fromlpinto itself.

The problem of finding the norm of a lower triangular matrix on the sequence spaceslp andlp(w) has been studied before in [–]. In the study, we will expand this problem for matrix operators fromlp(w) intocp(w) and matrix operators fromcp(w) intolp(w), and we consider certain matrix operators such as Cesàro, Nörlund and weighted mean. The study is an extension of some results obtained by [, ].

2 The norm of matrix operators fromlp(w)intocp(w)

In this section, we tend to compute the bounds for the norm of lower triangular matrix operators fromlp(w) intocp(w). In particular, we apply our results for lower triangular matrix operators fromlpintocp, whenwn=  for alln.

Throughout this paper, let A= (an,k) be a matrix with non-negative real entries i.e., an,k≥, for alln,k. This implies thatAxp,w,c≤ A|x|p,w,c, and hence the non-negative sequences are sufficient to determine the norm ofA. We say thatA= (an,k) is lower trian-gular ifan,k=  forn<k. A non-negative lower triangular matrix is called a summability matrix ifn_k=an,k=  for alln.

Lemma .([], Lemma .) Let a and x be two non-negative sequences,then for all n,

n

k= akxk≤

max ≤k≤n

 n–k+ 

n

j=k xj

_n

k=

(n–k+ )(ak–ak–)+_.

Lemma .([], Lemma .) Let N≥,and let a and x be two non-negative sequences.If xN ≥xN+≥ · · · ≥and xn= for n<N,then

n

k= akxk≥

 n

n

j= xj

naN+ n

n–N+  n

k=N+

(n–k+ )(ak–ak–)–

for all n.

Lemma .([], Lemma .) Let p> and w= (wn)be a decreasing sequence with non-negative entries and∞_n=wn

n be divergent.Let N≥and the matrix CN = (c N

n,k)be with the following entries:

cN_n,k= ⎧ ⎨ ⎩



n+N– for n≥k,  for n<k.

ThenCNp,wis determined by non-negative decreasing sequences andCNp,w=p∗. Note thatCis the well-known Cesàro matrix.

Lemma .([], Lemma .) If p> and x and w are two non-negative sequences and also w is decreasing,then

∞

j= wjmax

≤i≤j

 j–i+ 

j

k=i xk

p

≤p∗p

∞

k= wkxp_k.

We seta,=  andan,=  forn≥ and

MA=sup n≥

_n

k=

n–k+  n

_n

i=k ai,k–

n

i=k– ai,k–

+ ,

mA=sup N≥

inf n≥N

_n

i=N

ai,N+  n–N+ 

n

k=N+

(n–k+ ) _n

i=k ai,k–

n

i=k– ai,k–

– .

We are now ready to present the main result of this section.

Theorem . Suppose that p> and w= (wn)is a decreasing sequence with non-negative entries.If A= (an,k)is a lower triangular matrix with non-negative entries,then we have the following statements.

(i) Ap,w,c≤p∗MA.Moreover,ifMA<∞,thenAis a bounded matrix operator from lp(w)intocp(w).

(ii) If∞_n=wn

n is divergent and( wn

wn+)is decreasing,thenAp,w,c≥p

Therefore if w= (wn)is a decreasing sequence with non-negative entries and( wn wn+)is

decreasing and∞_n=wn

n =∞,then p∗mA≤ Ap,w,c≤p∗MA.

In particular,if wn= for all n and if MA<∞,then A is a bounded matrix operator from lpinto cpand p∗mA≤ Ap,c≤p∗MA.

Proof (i) Let (xn) be a non-negative sequence. By using Lemma ., we get

n k=  n n i=k ai,k xk ≤ max ≤k≤n

 n–k+ 

n j=k xj _n k=

n–k+  n _n i=k ai,k– n i=k– ai,k– +

≤MAmax ≤k≤n

 n–k+ 

n j=k xj .

By applying Lemma ., we deduce that

∞ n= wn _n k=  n n i=k ai,k xk p

≤Mp_A

∞

n=

wn max ≤k≤n

 n–k+ 

n

j=k xj

p

≤p∗MAp

∞

k= wkxp_k.

(ii) We havemA=sup_N≥βN, where

βN=inf n≥N

_n

i=N

ai,N+  n–N+ 

n

k=N+

(n–k+ ) _n i=k ai,k– n i=k– ai,k– – .

LetN≥, so thatβN≥. Ify= (yn) is a decreasing sequence with non-negative entries andyp,w= , we setx=x=· · ·=xN–=  and

xn+N–=

wn wn+N–

/p yn

for alln≥. Soxp,w=yp,w= , and from Lemma . it follows that

Ap_p,w,c≥

∞ n= wn _n k=  n n i=k ai,k xk p

≥β_Np

∞ n= wn  n n j= xj p

=β_Np

∞ n= wn+N–  n+N– 

n

j= xj+N–

=β_Np

∞

n= wn+N–

 n+N– 

n

j=

wj wj+N–

/p yj

p

≥β_NpCNypp,w,.

By Lemma ., we conclude thatAp,w,c≥p∗βN, so

Ap,w,c≥p∗mA.

In what follows we assume thatw= (wn) is a decreasing sequence with non-negative en-tries and ( wn

wn+) is decreasing and

∞

n= wn

n =∞.

At first we bring a corollary of Theorem . for a lower triangular matrixA= (an,k). The rows ofCAare increasing, whereCis the Cesàro matrix and

(CA)n,k=

∞

i=

c_n,iai,k=  n

n

i=k

ai,k, (n,k= , , . . .).

Corollary . Suppose that p> and A= (an,k)is a non-negative lower triangular matrix thatn_i=k–ai,k–≤_i=kn ai,kfor <k≤n.Then

Ap,w,c=p∗sup n≥

an,n.

In particular,Ip,w,c=p∗,where I is the identity matrix.

Proof Since the finite sequence (n_i=kai,k)n_k=is increasing for eachn, we have

_n

i=k ai,k–

n

i=k– ai,k–

+ =

n

i=k ai,k–

n

i=k– ai,k–

for ≤k≤n. Hence

MA=sup n≥

_n

k=

n–k+  n

_n

i=k ai,k–

n

i=k– ai,k–

=sup n≥  n

n

k= n

i=k

ai,k≤sup n≥

an,n.

Moreover, _n

i=k ai,k–

n

i=k– ai,k–

–

=  (≤k≤n)

and

mA=sup N≥

inf n≥N

n

i=N

ai,N=sup n≥

an,n.

Example . LetA= (an,k) be defined by

an,k= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 

n fork<n,

n–

n fork=n,  fork>n.

Since the finite sequence (n_i=kai,k)n

k= is increasing for eachn andsupn≥an,n= , by Corollary ., we haveAp,w,c= p∗.

Now, in the second case, we state some corollaries of Theorem . for a lower triangular matrixA, where the rows ofCAare decreasing.

Corollary . Suppose that p >  and A= (an,k) is a lower triangular matrix with n

i=k–ai,k–≥ n

i=kai,kfor <k≤n.Then

p∗

inf n≥  n

n

k= n

i=k ai,k

≤ Ap,w,c≤p∗

sup n≥

n

i= ai,

.

In particular,for summability matrices the left-hand side of the above inequality reduces to p∗.

Moreover,if the right-hand side of the above inequality is finite, then A is a bounded matrix operator from lp(w)into cp(w).

Proof Since the finite sequence (n_i=kai,k)n

k=is decreasing for eachn, we have _n

i=k ai,k–

n

i=k– ai,k–

+

=  ( <k≤n),

and (n_i=ai,–n_i=ai,)+₌n

i=ai,. HenceMA=supn≥ n

i=ai,. Moreover, _n

i=k ai,k–

n

i=k– ai,k–

– =

n

i=k ai,k–

n

i=k– ai,k–,

for  <k≤n, so

mA =sup N≥

inf n≥N

_n

i=N

ai,N+  n–N+ 

n

k=N+

(n–k+ ) _n

i=k ai,k–

n

i=k– ai,k–

=sup N≥

inf n≥N

 n–N+ 

n

k=N n

i=k ai,k

≥inf n≥  n

n

k= n

i=k ai,k.

Therefore, by Theorem ., we prove the desired result.

Example . Suppose thatα≥ and the matrixA= (an,k) is defined by

an,k= ⎧ ⎨ ⎩ 

nα forn≥k,

 forn<k.

Sincen_i=kai,k=n_i=kiα and

n

i=k–ai,k–≥ n

i=kai,kfor  <k≤n, we have ≤ Ap,w,c≤ p∗ζ(α), whereζ(α) =∞_n= 

nα.

Example . Suppose that the matrixA= (an,k) is defined by

an,k= ⎧ ⎨ ⎩



n(n+) forn≥k,  forn<k.

Sincen_i=kai,k=n_i=ki(i+) , by Corollary ., we have ≤ Ap,w,c≤p∗. We apply the above corollary to the following two special cases.

Let (an) be a non-negative sequence witha> , andAn=a+· · ·+an. The Nörlund matrixNa= (an,k) is defined as follows:

an,k= ⎧ ⎨ ⎩

an–k+

An for ≤k≤n,  fork>n.

Also the weighted mean matrixMa= (an,k) is defined by

an,k= ⎧ ⎨ ⎩ ak

An for ≤k≤n,  fork>n.

Corollary . Suppose that p> and Na= (an,k)is the Nörlund matrix and(an)is an increasing sequence.Then

p∗≤ Nap,w,c≤p∗

sup n≥

n

i= ai Ai

.

Proof SinceNais a summability matrix andn_i=ai,=n_i= ai

Ai, by applying Corollary .,

we have the desired result.

Corollary . Suppose that p> and Ma= (an,k)is the weighted mean matrix and(an) is a decreasing sequence.Then

p∗≤ Map,w,c≤p∗a

sup n≥

n

i=  Ai

.

Proof SinceMais a summability matrix and n

i=ai,= n

i= a

Ai, by Corollary ., the proof

is obvious.

Theorem . Suppose that p> and A= (an,k)is a non-negative lower triangular matrix. If A is a bounded matrix operator from lp(w)into itself,then A is a bounded matrix operator from lp(w)into cp(w)and

Ap,w,c≤p∗Ap,w.

Proof We have

Axp_p,w,c=

∞

n= wn

 n

n

k= k

j= ak,jxj

p

=

∞

n= wn

n

j=

(CA)n,jxj p

=(CA)x p p,w.

Hence, by Lemma ., we conclude thatAp,w,c=CAp,w≤p∗Ap,w. We apply the above theorem to the following two Nörlund and weighted mean matrices.

Corollary .([], Corollary .) Suppose that p> and Na= (an,k)is the Nörlund ma-trix and(an)is a decreasing sequence with an↓αandα> .Then

Nap,w=p∗.

Corollary . Suppose that p>  and Na= (an,k)is the Nörlund matrix and(an)is a decreasing sequence with an↓αandα> .Then

Nap,w,c≤

p∗.

Proof By applying Theorem . and Corollary ., we have the desired result.

Corollary .([], Corollary .) Suppose that p> and Ma= (an,k)is the weighted mean matrix and(an)is an increasing sequence with an↑αandα<∞.Then

Map,w=p∗.

Corollary . Suppose that p> and Ma= (an,k)is the weighted mean matrix and(an) is an increasing sequence with an↑αandα<∞.Then

Map,w,c≤

p∗.

Proof By using Theorem . and Corollary ., the proof is clear.

3 The norm of matrix operators fromcp(w)intolp(w)

In this section, we compute the bounds for the norm of lower triangular matrix operators fromcp(w) intolp(w). In particular, whenwn=  for alln, the bounds for the norm of lower triangular matrix operators fromcpintolp are deduced. Moreover, we apply our results for Cesàro, Nörlund and weighted mean matrices.

Proposition . ([],Proposition.).Let p> and w= (wn)be a decreasing sequence with non-negative entries,and let Cbe the Cesàro matrix.Then we haveCp,w≤p∗. Theorem . Suppose that p> and w= (wn)is a sequence with non-negative entries and

A= (an,k)is a lower triangular matrix with non-negative entries.We have



p∗Ap,w≤ Ac,p,w≤supn≥

n sup ≤k≤n

an,k

.

Moreover,if the right-hand side of the above inequality is finite,then A is a bounded matrix operator from cp(w)into lp(w).In particular,if wn= for all n,then we have



p∗Ap≤ Ac,p≤supn≥

n sup ≤k≤n

an,k

.

Proof Suppose thatx∈cp(w)

Axp_p,w=

∞

n= wn

n

k= an,kxk

p

≤

∞

n= wn

sup ≤k≤n

an,k n

k= xk

p

≤sup n≥

n sup ≤k≤n

an,k p∞

n= wn

 n

n

k= xk

p

=sup n≥

n sup ≤k≤n

an,k p

xp_p,w,c.

Hence

Axp,w

xp,w,c ≤ sup

n≥

n sup ≤k≤n

an,k

and

Ac,p,w≤sup n≥

n sup ≤k≤n

an,k

.

On the other hand, Proposition . concludes thatxp,w,c=Cxp,w≤p∗xp,w, so

Axp,w

xp,w,c ≥  p∗

Axp,w

xp,w .

Therefore_p∗Ap,w≤ Ac,p,w, and the proof is complete. Corollary . If p> ,then the generalized Cesàro matrix CN is bounded from cp(w)into lp(w)and

Proof Since

sup n≥

n sup ≤k≤n

an,k

=sup n≥

n n+N– = ,

by using Lemma . and Theorem ., the proof is obvious.

We apply the above theorem to the following two special cases.

Corollary . Suppose that p> and Na= (an,k)is the Nörlund matrix and(an)is a de-creasing sequence with an↓αandα> .Then

≤ Nac,p,w≤asup n≥

n An.

Proof By Theorem . and Corollary ., the proof is clear.

Example . Letα>  andan=α+_nα+ for alln. The sequence (an) is decreasing and

an↓αand alsoasup_n≥An n =  +



α. So

≤ Nac,p,w≤ + 

α.

SpeciallyNac,p,w→, whenα→ ∞.

Corollary . Suppose that p> and Ma= (an,k)is the weighted mean matrix and(an)is an increasing sequence with an↑αandα<∞.Then

≤ Mac,p,w≤sup n≥

nan An.

Proof By using Theorem . and Corollary ., the proof is obvious.

Example . Letan=  –_(n+)  for alln. The sequence (an) is increasing andan↑ and

also

sup n≥

nan An =

a

A .. So

≤ Mac,p,w≤..

4 Conclusions

In the present study, we considered the problem of finding bounds for the norm of lower triangular matrix operators fromlp(w) intocp(w) and fromcp(w) intolp(w). Moreover, we computed the norms of certain matrix operators such as Cesàro, Nörlund and weighted mean, and we extended some results of [, ].

Competing interests

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

The second author would like to record his pleasure to Dr. Gholareza Talebi [Department of Mathematics, Faculty of Sciences, Vali-E-Asr University of Rafsanjan] for his valuable conversations during the preparation of the present paper.

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Received: 22 December 2016 Accepted: 22 March 2017 References

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