Generalization of the space \(l(p)\) derived by absolute Euler summability and matrix operators

R E S E A R C H

Open Access

Generalization of the space

l(p)

derived by

absolute Euler summability and matrix

operators

Fadime Gökçe

_{and Mehmet Ali Sarıgöl}

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

University of Pamukkale, Denizli, Turkey

Abstract

The sequence spacel(p) having an important role in summability theory was defined and studied by Maddox (Q. J. Math. 18:345–355,1967). In the present paper, we generalize the spacel(p) to the space|Er

φ|(p) derived by the absolute summability of

Euler mean. Also, we show that it is a paranormed space and linearly isomorphic to

l(p). Further, we determine

α

β

γ

MSC: 40C05; 40F05; 46A45

Keywords: Absolute summability; Euler means; Matrix transformations; Sequences spaces

1 Introduction

LetX,Y be any subsets ofω, the set of all sequences of complex numbers, andA= (anv) be an infinite matrix of complex numbers. ByA(x) = (An(x)), we indicate theA-transform of a sequencex= (xv) if the series

An(x) =

∞

v=0 anvxv

are convergent forn≥0. IfAx∈Y, wheneverx∈X, thenA, denoted byA:X→Y, is called a matrix transformation fromXintoY, and we mean the class of all infinite ma-tricesAsuch thatA:X→Y by (X,Y). For cs,bs, andlp (p≥1), we write the space of all convergent, bounded,p-absolutely convergent series, respectively. Further, the matrix domain of an infinite matrixAin a sequence spaceXis defined by

XA=

x= (xn)∈ω:A(x)∈X

. (1)

Theα-,β-, andγ-duals of the spaceXare defined as follows:

Xα=∈ω: (nxn)∈l1for allx∈X

,

Xβ=∈ω: (nxn)∈csfor allx∈X

,

Xγ=∈ω: (nxn)∈bsfor allx∈X

.

A subspaceXis called anFKspace if it is a Frechet space, that is, a complete locally convex linear metric space, with continuous coordinatesPn:X→C(n= 1, 2, . . .), wherePn(x) =xn for allx∈X; anFKspace whose metric is given by a norm is said to be aBKspace. AnFK

spaceXincluding the set of all finite sequences is said to haveAKif

lim

m→∞x

[m]_{= lim}

m→∞ m

v=0

xve(v)=x

for every sequencex∈X, wheree(v)_{is a sequence whose only non-zero term is one invth}

place forv≥0. For example, it is well known that the Maddox space

l(p) =

x= (xn) :

∞

n=1

|xn|pn<∞

is anFKspace withAKwith respect to its natural paranorm

g(x) = _∞

n=0

|xn|pn 1/M

,

whereM=max{1,sup_npn}; also it is even aBK space ifpn≥1 for allnwith respect to the norm

x=inf

δ> 0 :

∞

n=0

|xn/δ|pn≤1

([19–21,29]).

Throughout this paper, we assume that 0 <infpn≤H<∞andp∗nis a conjugate ofpn,

i.e., 1/pn+ 1/p∗n= 1,pn> 1, and 1/p∗n= 0 forpn= 1.

Let avbe a given infinite series withsnas itsnth partial sum,φ= (φn) be a sequence of positive real numbers andp= (pn) be a bounded sequence of positive real numbers. The series avis said to be summable|A,φn|(p) if (see [10])

∞

n=1

(φn)pn–1An(s) –An–1(s)pn<∞.

It should be noted that the summability|A,φn|(p) includes some well-known summa-bility methods for special cases ofA,φ andp= (pn). For example, if we takeA=Erand

pn=kfor alln, then it is reduced to the summability method|E,r|k(see [12]) where Euler matrixEr_{is defined by}

er_nk= ⎧ ⎨ ⎩ _n

k

(1 –r)n–k_rk_{, 0≤k≤n,} 0, k>n,

for 0 <r< 1 and

e1_nk= ⎧ ⎨ ⎩

Also we refer the readers to the papers [7,9,30,31,35] for detailed terminology. A large literature body, concerned with producing sequence spaces by means of matrix domain of a special limitation method and studying their algebraic, topological structure and matrix transformations, has recently grown. In this context, the sequence spacesl(p),

rt

p,l(u,v,p), andl(Nt,p) were studied by Choudhary and Mishra [8], Altay and Başar [2,3], Yeşilkayagil and Başar [37] by defining as the domains of the band, Riesz, the factorable, and Nörlund matrices in thel(p) (see also [1,4–6,16–18,23–28]).

Also, some series spaces have been derived and examined by various absolute summabil-ity methods from a different point of view (see [13,14,32,34]). In this paper, we generalize the spacel(p) to the space|Er

φ|(p) derived by the absolute summability of Euler means and

show that it is a paranormed space linearly isomorphic tol(p). Further, we determineα-,

β-, andγ-duals of this space and construct its Schauder basis. Finally, we characterize certain matrix transformations on the space.

First, we remind some well-known lemmas which play important roles in our research.

2 Needed lemmas

Lemma 2.1([11]) Let p= (pv)and q= (qv)be any two bounded sequences of strictly positive

numbers.

(i) Ifpv> 1for allv,thenA∈(l(p),l1)if and only if there exists an integerM> 1such that

sup

_∞

v=0

n∈K

anvM–1 p

∗

v

:K⊂Nfinite

<∞. (2)

(ii) Ifpv≤1andqv≥1for allv∈N,thenA∈(l(p),l(q))if and only if there exists some

Msuch that

sup

v

∞

n=0

anvM–1/pv qn

<∞.

(iii) Ifpv≤1,thenA∈(l(p),c)if and only if

(a) lim

n anvexists for eachv, (b) sup_n,v |anv|

pv_<∞_,

andA∈(l(p),l∞)iff(b)holds.

(iv) Ifpv> 1for allv,thenA∈(l(p),c)if and only if(a) (a)holds,and(b)there is a

numberM> 1such that

sup

n

∞

v=0

anvM–1p ∗

v_<∞,

andA∈(l(p),l∞)iff(b)holds.

Lemma 2.2([33]) Let A= (anv)be an infinite matrix with complex numbers and(pv)be a

bounded sequence of positive numbers.If Up[A] <∞or Lp[A] <∞,then

(2C)–2Up[A]≤Lp[A]≤Up[A],

where C=max{1, 2H–1_},H=sup

vpv,

Up[A] =

∞

v=0

_∞

n=0

|anv| pv

and

Lp[A] =sup _∞

v=0

n∈K

anv

pv:K⊂N finite

.

Lemma 2.3([22]) Let X be an FK space with AK,T be a triangle,S be its inverse,and Y be an arbitrary subset ofω.Then we have A∈(XT,Y)if and only ifA∈(X,Y)and V(n)∈(X,c)

for all n,where

ˆ

anv=

∞

j=v

anjsjv; n,v= 0, 1, . . . ,

and

v(_mvn)= ⎧ ⎨ ⎩

m

j=vanjsjv, 0≤v≤m, 0, v>m.

3 Main theorems

In this section, we introduce the paranormed series space|Erφ|(p) as the set of all series

summable by the absolute summability method of Euler matrix and show that this space is linearly isomorphic to the spacel(p). Also, we compute the Schauder base,α-,β-, andγ -duals of the space and characterize certain matrix transformations defined on that space. First of all, we note that, by the definition of the summability|A,φn|(p), we can write the space|Er

φ|(p) as

E_φr(p) =

a∈ω:

∞

n=0

φpn–1

n Arn(s) pn

<∞

,

where

Ar_n(s) =Ar_n(s) –Ar_n–1(s)

and

Ar_n(s) = n

k=0

n k

Also, a few calculations give

Ar_n(s) = n

m=0

n

k=m

n k

(1 –r)n–krkam– n–1

m=0

n–1

k=m

n– 1

k

(1 –r)n–1–krkam

= n m=1 n

k=m

(1 –r)n–1–k

n– 1

k– 1 –r n k

rkam

= n

m=1

σnmam,

where

σnm= ⎧ ⎨ ⎩

n

k=m(1 –r)n–1–krk[ n–1

k–1

–rn_k], 1≤m≤n, 0, m>n.

Further, it follows by puttingr=q(1 +q)–1

σnm= (1 +q)1–n n

k=m

qk

n– 1

k– 1

–q(1 +q)–1

n k

= (1 +q)–n n

k=m

qk

n– 1

k– 1

–qk+1

n– 1

k

=qm(1 +q)–n

n– 1

m– 1

=

n– 1

m– 1

(1 –r)n–mrm.

Now, by considering Tr

n(φ,p)(a) =φ

1/p∗n

n Arn(s), we immediately get thatT0r(φ,p)(a) = a0φ

1/p∗₀

0 and

T_nr(φ,p)(a) =φ1/p∗n

n n

k=1

n– 1

k– 1

(1 –r)n–krkak

= n

k=1

t_nkr (φ,p)ak, (3)

where

t_nkr (φ,p) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

φ1/p∗0

0 , k=n= 0,

φ1/p∗n

n _n–1

k–1

(1 –r)n–krk, 1≤k≤n, 0, k>n.

(4)

Therefore, we can state the space|Erφ|(p) as follows:

Eφr(p) =

a= (ak) :

∞ n=1 φ

1/p∗n

n n

k=1

n– 1

k– 1

or

Eφr(p) =

l(p)_Tr_(φ,p)

according to notation (1).

Further, since every triangle matrix has a unique inverse which is a triangle (see [36]), the matrixTr_{(φ,p) has a unique inverseS}r_{(φ,p) = (s}r

nk(φ,p)) given by

sr_nk(φ,p) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

φ–1/p

∗

0

0 , k=n= 0,

φ–1/p

∗

k

k n–1

k–1

(r– 1)n–k_r–n_{, 1≤k≤n,} 0, k>n.

(5)

Before main theorems, note that ifr= 1 and φn= 1 for alln≥0, the space|Erφ|(p) is

reduced to the spacel(p).

Theorem 3.1 Let0 <r< 1and p= (pn)be a bounded sequence of non-negative numbers.

Then:

(a) The set|Eφr|(p)becomes a linear space with the coordinate-wise addition and scalar

multiplication,and also it is anFK-space with respect to the paranorm

x|Er_φ|(p)=

_∞

n=0

T_nr(φ,p)(x)pn 1/M

,

whereM=max{1,suppn}.

(b) The space|Er

φ|(p)is linearly isomorphic to the spacel(p),i.e.,|Erφ|(p)∼=l(p).

(c) Define a sequence(b(nv))bySr((e(v))) = ( nv=0srnv(φ,p)e(v)).Then the sequence(b

(v)

n )is

the Schauder base of the space|Erφ|(p).

(d) The space|Er

φ|(p)is separable.

Proof (a) The first part is a routine verification, so it is omitted. SinceTr_{(φ,p) is a} trian-gle matrix andl(p) is anFK-space, it follows from Theorem 4.3.2 in [36] that|Er

φ|(p) =

[l(p)]Tr_(φ,p)is anFK-space.

(b) We should show that there exists a linear bijection between the spaces|Er

φ|(p) and l(p). Now, considerTr_{(φ,p) :|E}r

φ|(p)→l(p) given by (3). Since the matrix corresponding

this transformation is a triangle, it is obvious thatTr(φ,p) is a linear bijection. Further-more, sinceTr_{(φ,p)(x)∈l(p) forx∈ |E}r

φ|(p), we get

x|Er φ|(p)=

_∞

n=0

T_nr(φ,p)(x)pn 1/M

=Tr(φ,p)(x)_l(p).

So,Tr_{(φ,p) preserves the paranorm, which completes this part of the proof.} (c) Since the sequence (e(v)_{) is the Schauder base of the space l(p) and |E}r

φ|(p) =

[l(p)]Tr_(φ,p), it can be written from Theorem 2.3 in [15] that b(v) = (Sr(φ,p)(e(v))) is a

Schauder base of the space|Er

φ|(p).

Theorem 3.2 Let0 <r< 1.Define

Dr₁=

a∈ω:∃M> 1,

∞

v=0

_∞

n=v

M–1b(_nv)an p∗v

<∞

,

Dr₂=

a∈ω:∃M> 1,sup

v

M1/pv

∞

n=v

b(_nv)an<∞

,

Dr

3=

a∈ω:

∞

n=v

b(v)

n anconverges for each v

,

Dr₄=

a∈ω:∃M> 1,sup

n n v=1 n

k=v

b(_kv)akM–1 p∗v

<∞

,

Dr₅=

a∈ω:sup

n,v

n

k=v

b(_kv)ak pv <∞ .

(i) Ifpv> 1for allv,then Eφr(p)

α

=Dr₁, Eφr(p)

β

=Dr₄∩Dr₃, Erφ(p)

γ

=Dr₄.

(ii) Ifpv≤1for allv,then

_Er

φ(p)

α

=Dr₂, Eφr(p)

β

=Dr₅∩Dr₃, Erφ(p)

γ

=Dr₅.

Proof To avoid the repetition of a similar statement, we only calculateβ-duals of|Er

φ|(p).

(i) Let us recall thata∈ {|Er

φ|(p)}β if and only ifax∈cswheneverx∈ |Eφr|(p). Now, by

using (5), it can be obtained that

n

k=0

akxk=T0r(φ,p)(x)φ –1/p∗₀

0 a0+

n k=1 ak k v=1

φ–1/p∗v

v

k– 1

v– 1

(r– 1)k–vr–kT_vr(φ,p)(x)

=Tr

0(φ,p)(x)φ –1/p∗₀

0 a0+

n

v=1

φ–1/p∗v

v Tvr(φ,p)(x) n

k=v

ak

k– 1

v– 1

(r– 1)k–v_r–k

= n

v=0

dnvTvr(φ,p)(x),

whereD= (dnv) is defined by

dnv= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

φ–1/p∗0

0 a0, n=v= 0,

n k=vb

(v)

k ak, 1≤v≤n, 0, v>n.

SinceTr_{(φ,p)(x)∈l(p) wheneverx∈ |E}r

φ|(p),a∈ {|Erφ|(p)}

β_{if and only ifD∈(l(p),c). So,}

it follows from Lemma2.1thata∈Dr

4∩Dr3ifpv> 1 for allv, and alsoa∈Dr5∩Dr3ifpv≤1 for allv.

Theorem 3.3 Let A= (anv)be an infinite matrix of complex numbers, (φn)and(ψn)be

sequences of positive numbers, p= (pn)and q= (qn)be arbitrary bounded sequences of

positive numbers with pn≤1and qn≥1for all n.Further,let the matrixA be defined byˆ

ˆ

anv=

∞

j=v

anjb(jv)

and F=Tr_(ψ,q)Aˆ_{.Then A∈(|E}r

φ|(p),|Erψ|(q))if and only if there exists an integer M> 1 such that,for n= 0, 1, . . . ,

∞

k=v

b(_kv)ank converges for each v, (6)

sup

m,v

m

k=v

b(_kv)ank pv

<∞, (7)

and

sup

v

∞

n=0

M–1/pv_f

nv qn

<∞. (8)

Proof Suppose thatpv≤1,qv≥1 for allv. Note that|Eφr|(p) = [l(p)]Tr_(φ,p) and|E_ψr|(q) =

[l(q)]Tr_(ψ,q). By Lemma2.3,A∈(|E_φr|(p),|Er_ψ|(q)) if and only ifAˆ∈(l(p),|E_ψr|(q)) andV(n)∈

(l(p),c), where the matrixV(n)_{is defined by}

v(n)

mv= ⎧ ⎨ ⎩

m j=vb

(v)

j anj, 0≤v≤m, 0, v>m.

One can see that sinceAˆ(x)∈ |Er

ψ|(q) = [l(q)]Tr_(ψ,q) wheneverx∈l(p),Aˆ ∈(l(p),|Er_ψ|(q))

iffF=Tr_(ψ,q)Aˆ _{∈(l(p),l(q)). Now, applying Lemma2.1(ii) and (iii) to the matricesFand}

V(n)_{, it follows thatV}(n)_{∈(l(p),c) iff, forn= 0, 1, . . . , conditions (6) and (7) hold, andF∈}

(l(p),l(q)) iff there exists an integerMsuch that

sup

v

∞

n=0

M–1/pv_f

nvqn<∞,

which completes the proof. Theorem 3.4 Assume that A= (anv)is an infinite matrix of complex numbers and(φn), (ψn)are sequences of positive numbers. If p= (pn)is an arbitrary bounded sequence of

positive numbers such that pn> 1for all n,and H=Tr(ψ, 1)Aˆ,then A∈(|Erφ|(p),|Erψ|(1)) if and only if there exists an integer M> 1such that,for n= 0, 1, . . . ,

∞

k=v

b(_kv)ank converges for each v (9)

sup

n

∞

v=0

n

k=v

b(_kv)ankM–1 p∗v

and

∞

v=0

_∞

n=0

M–1hnv p∗v

<∞. (11)

Proof Letpn> 1 for all n. It is clear that|Eφr|(p) = [l(p)]Tr_(φ,p) and|E_ψr|(1) =l_Tr_(ψ,1). So,

by Lemma2.3, we haveA∈(|Eφr|(p),|Erψ|(1)) if and only ifAˆ ∈(l(p),|Eψr|(1)) andV(n)∈

(l(p),c), whereAˆ andV(n) _{are given in Theorem3.3. If we takeH=T}r_{(ψ, 1)A}ˆ_{, then it is} easily seen thatAˆ ∈(l(p),|Er

ψ|(1)) iffH∈(l(p),l1) because, ifAˆ(x)∈ |Erψ|(1) for allx∈l1(p), H(x) =Tr_{(ψ, 1)(A}ˆ_(x))∈l

1. So, applying Lemma2.1(iv) to the matrixV(n), it is obtained

thatV(n)_{∈(l(p),c) iff conditions (9) and (10) are satisfied. Again, if we apply Lemma2.1(i)}

and Lemma2.2to the matrixH, then we haveH∈(l(p),l1) iff the last condition holds. 4 Conclusion

The sequence spaces defined as domains of Riesz, factorable, Nörlund andS-matrices in the spacesl(p) and the space of series summable by the absolute Euler have been recently studied by several authors. In this paper, we have defined the new absolute Euler space |Erφ|(p) and investigated some topological and algebraic properties such as isomorphism,

duals, base, and also characterized certain matrix transformations on that space. So, we have extended some well-known results.

Acknowledgements

We thank the editor and referees for their careful reading, valuable suggestions and remarks.

Funding

No funding was received.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript, read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 9 March 2018 Accepted: 7 June 2018

References

1. Alotaibi, A., Mursaleen, M., Alamri, B.A.S., Mohiuddine, S.A.: Compact operators on some Fibonacci difference sequence spaces. J. Inequal. Appl.2015, 203 (2015)

2. Altay, B., Ba¸sar, F.: On the paranormed Riesz sequence spaces of non-absolute type. Southeast Asian Bull. Math.26(5), 701–715 (2002)

3. Altay, B., Ba¸sar, F.: Generalization of the sequence spacel(p) derived by weighted mean. J. Math. Anal. Appl.330(1), 174–185 (2007)

4. Altay, B., Ba¸sar, F., Mursaleen, M.: On the Euler sequence spaces which include the spaceslpandl∞I. Inf. Sci.176(10), 1450–1462 (2005)

5. Altay, B., Ba¸sar, F., Mursaleen, M.: On the Euler sequence spaces which include the spaceslpandl∞II. Nonlinear Anal. 65(3), 707–717 (2006)

6. Ba¸sarır, M., Kara, E.E., Konca, ¸S.: On some new weighted Euler sequence spaces and compact operators. Math. Inequal. Appl.17(2), 649–664 (2014)

7. Bor, H.: On|N,pn|ksummability factors of infinite series. Tamkang J. Math.16(1), 13–20 (1985) 8. Choudhary, B., Mishra, S.K.: A note on Köthe–Toeplitz duals of certain sequence spaces and their matrix

transformations. Int. J. Math. Math. Sci.18(4), 681–688 (1995)

9. Flett, T.M.: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math. Soc.7, 113–141 (1957)

10. Gökçe, F., Sarıgöl, M.A.: Matrix operators on the series space|Nθp|(μ) and applications. Kuwait J. Sci. (in press) 11. Grosse-Erdmann, K.G.: Matrix transformations between the sequence spaces of Maddox. J. Math. Anal. Appl.180,

12. Hardy, G.H.: Divergent Series, vol. 334. Am. Math. Soc., Providence (2000)

13. Hazar, G.C., Gökçe, F.: Characterizations of matrix transformations on the series spaces derived by absolute factorable summability. In: Developments in Science and Engineering, pp. 411–426 (2016)

14. Hazar, G.C., Sarıgöl, M.A.: Compact and matrix operators on the space|C, –1|k. J. Comput. Anal. Appl.25, 1014–1024 (2018)

15. Jarrah, A.M., Malkowsky, E.: Ordinary absolute and strong summability and matrix transformations. Filomat17, 59–78 (2003)

16. Kara, E.E., Ba¸sarır, M.: On compact operators and some EulerB(m)_{-difference sequence spaces. J. Math. Anal. Appl.}

279(2), 459–511 (2011)

17. Kara, E.E., Ilkhan, M.: On some Banach sequence spaces derived by a new band matrix. Br. J. Math. Comput. Sci.9(2), 141–159 (2015)

18. Kara, E.E., Ilkhan, M.: Some properties of generalized Fibonacci sequence spaces. Linear Multilinear Algebra64(11), 2208–2223 (2016)

19. Maddox, I.J.: Spaces of strongly summable sequences. Q. J. Math.18, 345–355 (1967)

20. Maddox, I.J.: Paranormed sequence spaces generated by infinite matrices. Math. Proc. Camb. Philos. Soc.64, 335–340 (1968)

21. Maddox, I.J.: Some properties of paranormed sequence spaces. J. Lond. Math. Soc. (2)1, 316–322 (1969) 22. Malkowsky, E., Rakocevic, V.: On matrix domains of triangles. Appl. Math. Comput.189(2), 1146–1163 (2007) 23. Mohiuddine, S.A.: Matrix transformations of paranormed sequence spaces through de la Vallée–Pousin mean. Acta

Sci., Technol.37(1), 71–75 (2015)

24. Mohiuddine, S.A., Hazarika, B.: Some classes of ideal convergent sequences and generalized difference matrix operator. Filomat31(6), 1827–1834 (2017)

25. Mohiuddine, S.A., Raj, K.: Vector valued Orlicz–Lorentz sequence spaces and their operator ideals. J. Nonlinear Sci. Appl.10, 338–353 (2017)

26. Mohiuddine, S.A., Raj, K., Alotaibi, A.: Generalized spaces of double sequences for Orlicz functions and bounded-regular matrices overn-normed spaces. J. Inequal. Appl.2014, 332 (2014)

27. Mursaleen, M., Mohiuddine, S.A.: Regularlyσ-conservative andσ-coercive four dimensional matrices. Comput. Math. Appl.56(6), 1580–1586 (2008)

28. Mursaleen, M., Mohiuddine, S.A.: Onσ-conservative and boundedlyσ-conservative four-dimensional matrices. Comput. Math. Appl.59(2), 880–885 (2010)

29. Nakano, H.: Modulared sequence space. Proc. Jpn. Acad., Ser. A, Math. Sci.27, 508–512 (1951) 30. Sarıgöl, M.A.: On absolute summability factors. Comment. Math. Prace Mat.31, 157–163 (1991) 31. Sarıgöl, M.A.: On local properties of factored Fourier series. Appl. Math. Comput.216, 3386–3390 (2010)

32. Sarıgöl, M.A.: Matrix transformations on fields of absolute weighted mean summability. Studia Sci. Math. Hung.48(3), 331–341 (2011)

33. Sarıgöl, M.A.: An inequality for matrix operators and its applications. J. Class. Anal.2, 145–150 (2013)

34. Sarıgöl, M.A.: Spaces of series summable by absolute Cesàro and matrix operators. Commun. Math. Appl.7(1), 11–22 (2016)

35. Sulaiman, W.T.: On summability factors of infinite series. Proc. Am. Math. Soc.115, 313–317 (1992)

36. Wilansky, A.: Summability Through Functional Analysis. Mathematics Studies, vol. 85. North-Holland, Amsterdam (1984)