# A New RH-Regular Matrix Derived by Jordan’s Function and Its Domains on Some Double Sequence Spaces

Authors: Sezer Erdem; Serkan Demiriz
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In the present study, we introduce a new RH-regular 4D (4-dimensional) matrix derived by Jordan’s function and define double sequence spaces by using domains of 4D Jordan totient matrixJt on some classical double sequence spaces. Also, the α-, βϑ-, and γ-duals of these spaces are determined. Finally, some classes of 4D matrices on these spaces are characterized.


---

## Body

## 1. Introduction and Preliminaries

The Jordans’s functionJt:ℕ⟶ℕ, k⟼Jtk is described as the number of t-tuples of positive integers all less than or equal to k that form a coprime with t+1-tuples together with k, where k,t∈ℕ and ℕ=1,2,⋯. Jt is entitled as Jordan’s function after Camille Jordan, and it is one of many generalizations of the famous Euler-totient function φ. Since the equality Jtk1k2=Jtk1Jtk2 holds for the coprime numbers k1,k2∈ℕ, Jt is multiplicative, and furthermore, it is an arithmetic function. If a1b1a2b2a3b3⋯aibi is the prime factorization of k∈ℕ for k>1, in that case,(1)Jtk=kt1−1a1t1−1a2t1−1a3t⋯1−1ait.It is also significant to say that the Euler-totient functionφ is the special case of the Jordan’s function Jt for t=1. Other three important features of Jordan’s function are(2)∑k∣mJtk=mt,∑k∣mμkkt=Jtmmt,and(3)∑k∣mμmkkt=Jtm,where μ denotes the Möbius function (see [1]). Given any k∈ℕ, the function μ is described as(4)μk≔1k=1,−1i,k=a1a2⋯ai,wherea1,a2,⋯,aiaredifferentprimenumbers,0,a2∣kforatleastoneprimenumbera.Ifa1b1a2b2a3b3⋯aibi is the prime factorization of k∈ℕ such that k>1, in this fact, ∑k∣mkμk=1−a11−a21−a3⋯1−ai. For m≠1, the equality ∑k∣mμk=0 holds and μk1k2=μk1μk2, where k1,k2∈ℕ are coprime.A double sequence is a function such thatf:ℕ×ℕ⟶ζ, k,l⟼fk,l=xkl, where ζ denotes any nonempty set. Any linear subspace of the linear space Ω≔x=xkl:xkl∈ℂ,k,l∈ℕ is named as double sequence space. Here, the symbol ℂ represents the complex field. x=xkl∈Ω is convergent in the Pringsheim’s sense provided that for all ξ>0, there subsists nξ∈ℕ with xkl−κ<ξ for k,l>nξ. In that case, κ∈ℂ is named as the Pringsheim limit of x and stated with p−limk,l⟶∞xkl=κ. Cp represents the space of all such x which are called shortly as p-convergent. The space of all bounded double sequences is denoted by Mu. Of particular interest is unlike single sequences, when p−limk,l⟶∞xkl=κ, it might be x∉Mu. The bounded sequences which are also p-convergent are indicated by Cbp, that is, Cbp=Cp∩Mu. x=xkl∈Cp is entitled as regularly convergent if the limits xk≔limlxkl, k∈ℕ and xl≔limkxkl, l∈ℕ subsist, and the limits limklimlxkl and limllimkxkl subsist and are equivalent to the p−limx. The set of all regularly convergent double sequences is represented by Cr, and also, Mu, Cbp, and Cr are Banach spaces with the norm x∞=supk,l∈ℕxkl. Moreover, CSϑ and BS symbolize the spaces of all ϑ-convergent and bounded series, respectively. It should be noted that the double sequence space Lu which was defined by Zeltser [2] is the special case of the space Ls described by(5)Ls≔xkl∈Ω:∑k,l=1∞,∞xkls<∞,0<s<∞,for s=1. It is also significant to advert that the space Ls is Banach space with(6)xLs=∑k,lxkls1/s.1≤s<∞.The concept of almost convergence which is more general than the convergence of single sequences was acquainted by Lorentz [3], and later, present idea was conveyed and examined for double sequences in [4]. The spaces of all almost convergent and almost null double sequences denoted by Cf and Cf0, respectively.Consider thatx∈Ω and describe Y=υmn as υmn≔∑k=1m∑l=1nxkl,m,n∈ℕ. In that case, the pair xmn,υmn is entitled as double series. Here, the sequence Y=υmn is the sequence of partial sums of the double series.Throughout this article, it will be used∑k,l instead of ∑k=1∞∑l=1∞, 4D instead of 4-dimensional, ϑ∈p,bp,r, 1/s+1/s′=11<s<∞, and m,n,k,l∈ℕ. Double sequences emn=eklmn, and e described by(7)eklmn≔1,ifm,n=k,l,0fortheothercases,and e=∑m,nem,n. If bmnkl=0 for k>m or l>n or both, B=bmnkl is named as triangular matrix, and also, if bmnmn≠0, in that case, B is called as triangle.Let us considerΨ,Λ⊂Ω and the 4D complex infinite matrix B=bmnkl. If for all x=xkl∈Ψ, Bxmn=ϑ−∑k,lbmnklxkl subsists and is in Λ for each m,n∈ℕ, in that case, B describes a matrix transformation from Ψ into Λ, and it is written as B:Ψ→Λ. Ψ:Λ stands for the class of all 4D complex infinite matrices from Ψ into Λ. In that case, B∈Ψ:Λ if and only if Bmn∈Ψβϑ and Bx∈Λ for all x∈Ψ, where Bmn=bmnklk,l∈ℕ and m,n∈ℕ.The domainΨBϑ of a 4D complex infinite matrix B in a double sequence space Ψ consists of the sequences whose B-transforms are in Ψ; that is,(8)ΨBϑ≔x=xkl∈Ω:Bx≔ϑ−∑klbmnklxklm,n∈ℕsubsistsandisinΨ.The 2D Jordan matrix and its domain on the space ofℓp of absolutely p-summable single sequences are described and examined by İlkhan et al. [5]. The 4D Euler-totient matrix Φ⋆ and domains of this matrix on double sequences Ls, Mu, Cp, Cbp, and Cr were described and examined by Demiriz and Erdem [6] and Erdem and Demiriz [7].Suppose thatRqt, E(r,s), B(r,s,t,u), B~, Φ⋆, and Br,s denote the 4D Riesz, Euler, generalized difference, sequential band, Euler-totient, and binomial matrices, respectively. The domains ΨB of some 4D triangle matrices B in a space Ψ are listed in Table 1.Table 1
Some 4D triangle matrices and their domains.
BΨΨBRefer to:RqtLsRqtLs[8]Er,sLp,Muεpr,s,ε∞r,s[9]Br,s,t,uMu,Cbp,Cp,Cr, LqBMu,BCbp,BCp,BCr,BLq[10]Br,s,t,uCf,Cf0BCf,BCf0[11]B~Mu,Cbp,Cp,Cr, LqB~Mu,B~Cbp,B~Cp,B~Cr, B~Lq[12]Φ⋆LpΦ⋆Lp[6]Φ⋆Mu,Cbp,Cp,CrΦ⋆Mu,Φ⋆Cbp,Φ⋆Cp, Φ⋆Cr[7]Br,sMu,Cp,Cbp,CrB∞r,s,Br,s,Bbpr,s,Bregr,s[13]Br,sCf,Cf0Bfr,s,Bf0r,s[14]


## 2. The 4D Jordan Totient Matrix and Some Double Sequences

In this section, we define 4D Jordan totient matrixJt derived by Jordan’s function, and we show that this matrix is RH-regular. By using its domains on the spaces Mu, Cp, Cbp, Cr, and Ls for 0<s<∞, we introduce the spaces J∞t, Jpt, Jbpt, Jrt, and Jst and also give inclusion relations related these newly described spaces. The reader can refer to the papers [15–30] and recent monograph [31] concerning with the domain of 4D triangles in the classical spaces of double sequences and related topics.Now, we describe the 4D Jordan totient matrixJt=jmnklt as follows:(9)jmnklt≔JtkJtlmntk∣m,l∣n,0,fortheothercases,for each t∈ℕ. It should be noted that the 4D Euler-totient matrix Φ⋆ defined by Demiriz and Erdem [6] is the special case of the matrix Jt for t=1. Moreover, it is clear that Jt is a triangle, and the Jt-transform of any x=xkl∈Ω is described as(10)ymn≔Jtxmn=1mnt∑km,lnJtkJtlxkl.The inverseJt−1=jtmnkl−1 of the triangle matrix Jt is calculated as(11)jtmnkl−1≔μm/kμn/lJtmJtnklt,k∣m,l∣n,0,fortheothercases.In that case,xmn is obtained by applying Jt−1 to (10) as(12)xmn=∑km,lnμm/kμn/lJtmJtnkltykl.Definition 1 (see [32, 33]).
For everyx∈Cbp, if Bx∈Cp and bp−limx=p−limBx, then B is called as RH-regular.Lemma 2 (see [32, 33]).
A 4D matrixB=bmnkl is RH-regular if and only if(13)RH1:p−limm,n⟶∞bmnkl=0k,l∈ℕ,RH2:p−limm,n⟶∞∑k,lbmnkl=1,RH3:p−limm,n⟶∞∑kbmnkl=0l∈ℕ,RH4:p−limm,n⟶∞∑lbmnkl=0k∈ℕ,RH5:∑k,lbmnklisp−convergent,RH6:∑k,l>Lbmnkl<KforatleastK,L∈ℕ.Theorem 3.
The 4D Jordan totient matrixJt described by (9) is RH-regular.Proof.
SinceJtk/mt⟶0 (m⟶∞) and Jtl/nt⟶0 (n⟶∞) from the regularity of the 2D Jordan totient matrix [5], we obtain that JtkJtl/mnt⟶0(m,n⟶∞). Thus, RH-1 holds. From the equality(14)∑k,lJtkJtlmnt=∑k∣m,l∣nJtkJtlmnt=1,we see that RH-2 holds. Since(15)∑kjmnklt=∑kJtkJtlmnt=Jtlnt∑k∣mJtkmt=Jtlnt,then p−limm,n⟶∞∑kjmnklt=0 for each l and RH-3 holds. It can be shown similar with RH-3 that the 4D Jordan totient matrix satisfies RH-4. From the relation ∑k,ljmnklt=1, we see that RH-5 holds. By using the relation ∑k,ljmnklt=1 and the positivity of the matrix Jt, it is clear that the condition RH6 satisfies.Now, we introduce the double sequence spaces by using the domains of 4D Jordan totient matrixJt as follows:(16)J∞t=x=xmn∈Ω:supm,n∈ℕJtxmn<∞,Jpt=x=xmn∈Ω:∃κ∈ℂ∋p−limm,n→∞Jtxmn−κ=0,Jbpt=x=xmn∈Ω:Jtx∈Cbp,Jrt=x=xmn∈Ω:Jtx∈Cr,Jst=x=xmn∈Ω:∑m,nJtxmns<∞.0<s<∞.The sets which are newly defined above can be rewritten asJ∞t=MuJt, Jϑt=CϑJt, and Jst=LsJt with the notation (8), where ϑ∈p,bp,r.Theorem 4.
The setsJ∞t, Jpt, Jbpt, Jrt, and Jst are linear spaces, and the following statements hold:(i)
The spacesJ∞t, Jbpt, and Jrt are Banach spaces with(17)xJ∞t=Jtx∞=supm,n∈ℕ1mnt∑km,lnJtkJtlxkl,which are linearly norm isomorphic to Mu, Cbp, and Cr, respectively.(ii)
The spaceJpt is a complete seminormed space with(18)xJpt=limi→∞supm,n≥iJtxmn,which is linearly isomorphic to the space Cp.(iii)
For0<s<1, Jst is a complete s-normed space with(19)xJst≀=JtxLs≀=∑m,n1mnt∑km,lnJtkJtlxkls,which is linearly s-norm isomorphic to the space Ls.(iv)
For1≤s<∞, Jst is a Banach space with(20)xJst=Jtxs=∑m,n1mnt∑km,lnJtkJtlxkls1/s,which is linearly norm isomorphic to the space Ls.Proof.
Since the other items of the theorem can be proven with similar technique, we will pass the others except first one for the setJ∞t.
It is left to the reader to show thatJ∞t is a normed linear space. To confirm the fact that J∞t is linearly norm isomorphic to Mu, it is needed to sure the presence of any linear and norm keeping bijection from J∞t to Mu. Let us describe the transformation L:J∞t⟶Mu such that Lx=Jtx for all x∈J∞t. It is easy to see the linearity of L. Furthermore, we can understand the injectivity of L from the fact that Lx=0⇒x=0. Let us suppose the sequences y=ykl∈Mu and x=xmn whose terms are(21)xmn=∑km,lnμm/kμn/lJtmJtnkltykl.
So, we get(22)Jtxmn=1mnt∑km,lnJtkJtlxkl=1mnt∑km,lnJtkJtl∑ik,rlμk/iμl/rJtkJtlirtyir=1mnt∑km,ln∑ik,rlμk/iμl/rirtyir=ymn,which yields us that supm,nJtxmn=supm,nymn. Thus, it is obtained that x∈J∞t, since y∈Mu. That is, L is surjective and since the equality xJ∞t=Jtx∞ holds, L keeps the norm. Moreover, Mu is a Banach space, and we obtain the desired result from Section (b) of Corollary 6.3.41 in [34].
Now, we may give two theorems and a lemma related the inclusion relations.Theorem 5.
Mu⊂J∞t.Proof.
Considerx∈Mu. Since,(23)xJ∞t=supm,n∈ℕ1mnt∑km,lnJtkJtlxkl≤x∞supm,n∈ℕ1mnt∑km,lnJtkJtl=x∞<∞,it is obtained that x∈J∞t, and thus, Mu is subspace of J∞t.
Now, we give a lemma which is needed to prove our results.Lemma 6 (see [35]).
Suppose thatB=bmnkl is a 4D infinite matrix. Then, B∈Cp:Cϑ if and only if following conditions hold:(24)supm,n∈ℕ∑k,lbmnkl<∞,(25)∃akl∈ℂ∋ϑ−limm,n⟶∞bmnkl=aklsubsists,(26)∃κ∈ℂ∋ϑ−limm,n⟶∞∑k,lbmnkl=κsubsists,(27)∀k∈ℕ,∃l0∋bmnkl=0,∀l>l0,(28)∀l∈ℕ,∃k0∋bmnkl=0,∀k>k0.Theorem 7.
Cp is the subspace of Jϑt, where ϑ∈p,bp,r.Proof.
To show the validity of the theorem, we just need to show that the 4D Jordan totient matrixJt satisfies the Lemma 6’s conditions.
From the equalitysupm,n∈ℕ∑k,ljmnklt=supm,n∈ℕ∑k,lJtkJtl/mnt=1, (24) holds. Since Jt is RH-regular, then (25) holds. From the equality ϑ−limm,n⟶∞∑k,ljmnklt=1, (26) holds. For all k∈ℕ, if we take l0=n, then jmnklt=0 for all l>l0 and m,n∈ℕ. Therefore, (27) holds. The condition (28) can be shown similarly with (27).


## 3. Duals

Duals of aΨ⊂Ω are described by the following way:(29)Ψα≔c=ckl∈Ω:∑k,lcklxkl<∞forallxkl∈Ψ,Ψβϑ≔c=ckl∈Ω:ϑ−∑k,lcklxklsubsistsforallxkl∈Ψ,Ψγ≔c=ckl∈Ω:supm,n∈ℕ∑k,l=1m,ncklxkl<∞forallxkl∈Ψ.Note thatΨα⊂Ψγ and if Ψ⊂Λ, then Λα⊂Ψα.It is listed the spaces and their duals which are computed in this section in Table2.Table 2
Spaces and their duals which are computed in this section.
J∞tJptJbptJrtJst0<s≤1Jst1<s<∞α,βp,βbp,γβϑα,βϑ,γβϑα,βbp,γβbp,γTheorem 8.
The following statements hold:(30)J∞tα=Lu,Jbptα=Lu.Proof.
(i)
Since it can be proven by applying a similar way used in Theorem 3.1 in [7], we omit details(ii)
For this part of the proof, it will be used contradiction method. Let us take a sequencec∈Jbptα\Lu. In that case, ∑m,ncmnxmn<∞ for all x∈Jbpt. Consider that e=∑m,nemn∈Jbpt. Thus, we obtain that ce=c∉Lu, that is c∉Jbptα. Therefore, it must be c∈Lu and consequently the inclusion Jbptα⊂Lu holds
Conversely, let us takec=cmn∈Lu and x∈Jbpt. In this fact, we understand that y∈Cbp from the equality (10), and therefore, supmn∣ymn∣≤K. Thus, we obtain from the following inequality(31)∑m,ncmnxmn=∑m,ncmn∑k∣m,l∣nμm/kμn/lJtmJtnkltykl≤K∑m,ncmn,that c∈Jbptα and this completes the proof.Let us state the following conditions which will be used to give the 4D martix classes (see [2, 8, 10, 22, 32, 35–37]).(32)supm,n,k,l∈ℕbmnkl<∞,supm,n∈ℕ∑k,lbmnkls′<∞,supk,l∈ℕ∑m,nbmnkls1<∞,∃k0∈ℕ∋ϑ−limm,n→∞∑lbmn,k0,l−ak0l=0,∃l0∈ℕ∋ϑ−limm,n→∞∑kbmnk,l0−ak,l0=0,∃l0∈ℕ∋ϑ−limm,n→∞∑kbmnk,l0=ρl0,∃k0∈ℕ∋ϑ−limm,n→∞∑lbmn,k0,l=ρk0,∃akl∈ℂ∋bp−limm,n→∞∑k,lbmnkl−akl=0,bp−limm,n→∞∑k=1mbmnklsubsists,∀l∈ℕ,bp−limm,n→∞∑l=1nbmnklsubsists,∀k∈ℕ,∑k,lbmnklconverges.It is listed necessary and sufficient conditions ofΨ:Λ with the following table.Theorem 9.
Consider the setF defined by(33)F=c=cmn∈Ω:supk,l∈ℕ∑m,nμm/kμn/lJtmJtnkltcmn<∞.
If0<s≤1, then Jstα=F.Proof.
Let us describe the 4D matrixG=gmnkl by(34)gmnkl≔μm/kμn/lJtmJtnkltcmn,k∣m,l∣n,0,fortheothercases.
In this fact, by using the equality (12), we obtain that(35)cmnxmn=cmn∑k∣m,l∣nμm/kμn/lJtmJtnkltykl=∑k∣m,l∣nμm/kμn/lJtmJtnkltcmnykl=Gymn.
Thus, we conclude from the relation (35) that cx∈Lu whenever x∈Jst if and only if Gy∈Lu whenever y∈Ls. This fact implies that c∈Jstα if and only if G∈Ls:Lu. Therefore, by using the condition of the class Ls:Ls1 for 0<s≤1 and s1=1 in Table 3 with the matrix G=gmnkl instead of B=bmnkl, we obtain the desired consequence.Theorem 10.
Let us takeΨ⊂Ω, c=cmn∈Ω and the 4D matrix O=omnkl as(36)omnkl≔∑a=k,k∣am∑b=l,l∣bnμa/kμb/lJtaJtbkltcab,1≤k≤m,1≤l≤n,0,otherwise.
In that case,(37)ΨJtβϑ=c=cmn∈Ω:O∈Ψ:Cϑ,ΨJtγ=c=cmn∈Ω:O∈Ψ:Mu.Proof.
(i)
Let us take sequencesc=cmn∈Ω and x∈ΨJt. In that case, y=ymn∈Ψ with Jtx=y. So, from the equality (12),(38)zmn=∑k,l=1m,ncklxkl=∑k,l=1m,nckl∑ak,blμk/aμl/bJtkJtlabtyab=∑k,l=1m,n∑a=k,k∣am∑b=l,l∣bnμa/kμb/lJtaJtbkltcabykl=Oymn,for all m,n∈ℕ. Then, by considering the equality above, we deduce that cx∈CSϑ whenever x∈ΨJt if and only if z∈Cϑ whenever y∈Ψ. This leads us to the fact that c∈ΨJtβϑ if and only if O∈Ψ:Cϑ, as claimed.(ii)
This part can be easily proven similarly to the first part by using the description of theγ-dual. So, we omit the proof.
Now, consider the setsϖf which are defined by(39)ϖf=c=cmn∈Ω:Conditionfholdswithomnklinsteadofbmnkl,where the 4D matrix O=omnkl defined as in Theorem 10 and 5≤f≤20.Theorem 11.
The following statements hold:(40)Jptβϑ=⋂f=59ϖf,Jbptβϑ=⋂f=57ϖf∩ϖ13∩ϖ14,Jrtβϑ=⋂f=57ϖf∩ϖ15∩ϖ16.Proof.
From Theorem10 and the conditions of the classes Cp:Cϑ, Cbp:Cϑ, and Cr:Cϑ in Table 3 with omnkl instead of bmnkl, the accuracy of items (i), (ii), and (iii) can be easily seen, respectively.Theorem 12.
The following statements hold:(41)J∞tβbp=ϖ5∩ϖ6⋂f=1720ϖf.(i)
For0<s≤1, Jstβbp=ϖ6∩ϖ10(ii)
For1<s<∞, Jstβbp=ϖ6∩ϖ11(42)J∞tβp=ϖ6∩ϖ8∩ϖ9.Proof.
From Theorem10 and the conditions of the classes Mu:Cbp, Ls:Cbp for 0<s≤1, Ls:Cbp for 1<s<∞ and Mu:Cp in Table 3 with omnkl instead of bmnkl, the accuracy of items (i), (ii), (iii), and (iv) can be easily seen, respectively.Theorem 13.
The following statements hold:(43)J∞tγ=ϖ5,Jbptγ=ϖ5.(i)
For0<s≤1, Jstγ=ϖ10(ii)
For1<s<∞, Jstγ=ϖ11Proof.
From Theorem10 and the conditions of the classes Mu:Mu, Cbp:Mu, Ls:Mu for 0<s≤1, and Ls:Mu for 1<s<∞ in Table 3 with omnkl instead of bmnkl, the accuracy of items (i), (ii), (iii), and (iv) can be easily seen, respectively.Table 3
Characterizations ofΨ:Λ, where Ψ,Λ∈Mu,Cp,Cbp,Cr,Ls.
(Ψ↓:Λ→)MuCpCbpCrLs11≤s1<∞Mu(5)(6), (8), (9)(5), (6), (17), (18), (19), (20)••Cp•(5), (6), (7), (8), (9)(5), (6), (7), (8), (9)(5), (6), (7), (8), (9)•Cbp(5)(5), (6), (7), (13), (14)(5), (6), (7), (13), (14)(5), (6), (7), (13), (14)•Cr•(5), (6) ,(7), (15), (16)(5), (6), (7), (15), (16)(5), (6), (7), (15), (16)•Ls0<s≤1(10)•(6), (10)•(12)Ls1<s<∞(11)•(6), (11)••The symbol • represents unknown conditions for (Ψ:Λ).


## 4. 4D Matrix Transformations

In the current part of the article, it will be given the classesΨJt:Λ and Ψ:ΛJt, where Ψ,Λ∈Mu,Cbp,Cp,Cr,Ls.Theorem 14.
Consider the 4D infinite matricesB=bmnkl and H=hmnkl whose elements are connected with the equality(44)hmnkl=∑a=k,k∣a∞∑b=l,l∣b∞μa/kμb/lJtaJtbkltbmnab.
In that case;B∈ΨJt:Λ if and only if Bmn∈ΨJtβϑ and H∈Ψ:Λ, where Ψ,Λ∈Mu,Cbp,Cp,Cr,Ls.Proof.
Assume thatB∈ΨJt:Λ. In that case, Bx exists and is in Λ for all x∈ΨJt such that y=Jtx∈Ψ, and it also implies that Bmn∈ΨJtβϑ. By bearing in mind (12), we obtain that(45)∑k,l=1i,jbmnklxkl=∑k,l=1i,jbmnkl∑a∣k,b∣lμk/aμl/bJtkJtlabtyab=∑k,l=1i,j∑a=k,k∣ai∑b=l,l∣bjμa/kμb/lJtaJtbkltbmnabykl,for all i,j∈ℕ. In that case, if we take ϑ-limit on the equality (45) as i,j⟶∞, we have Bx=Hy. Therefore, we obtain that Hy∈Λ whenever y∈Ψ, that is H∈Ψ:Λ.
Conversely, suppose thatBmn∈ΨJtβϑ and H∈Ψ:Λ. Let us take the sequence x∈ΨJt with y=Jtx∈Ψ. In that case, Bx exists. By using the relation (12), one can derive from the relation (45) when passing ϑ-limit as i,j⟶∞ that(46)∑k,lbmnklxkl=∑k,lhmnklykl,which implies that Bx=Hy. Therefore, B∈ΨJt:Λ, and this completes the proof.Corollary 15.
Consider the 4D infinite matricesB=bmnkl and H=hmnkl whose elements are connected with the relation (44). In that case, the necessary and sufficient conditions for the classes ΨJt:Λ can be read from the Table 4.Lemma 16 (see [8]).
LetΨ,Λ⊂Ω, B=bmnkl be a 4D matrix and F=fmnkl also be a 4D triangle. In that case, B∈Ψ:ΛF if and only if FB∈Ψ:Λ.
Now, let us define the 4D matrixU=umnkl by(47)umnkl=∑i∣m,d∣njmnidtbidkl,and give the following corollary.Corollary 17.
Consider the 4D infinite matricesB=bmnkl and U=umnkl whose elements are connected with the relation (47). In that case, the necessary and sufficient conditions for the classes Ψ:ΛJt can be read from Table 5.Table 4
Characterizations ofΨJt:Λ, where Ψ,Λ∈Mu,Cbp,Cp,Cr,Ls.
(ΨJt↓:Λ→)MuCpCbpCrLs11≤s1<∞J∞t(5)(6), (8), (9)(5), (6), (17), (18), (19), (20)••Jpt•(5), (6), (7), (8), (9)(5), (6), (7), (8), (9)(5), (6), (7), (8), (9)•Jbpt(5)(5), (6), (7), (13), (14)(5), (6), (7), (13), (14)(5), (6), (7), (13), (14)•Jrt•(5), (6), (7), (15), (16)(5), (6), (7), (15), (16)(5), (6), (7), (15), (16)•Jst0<s≤1(10)•(6), (10)•(12)Jst1<s<∞(11)•(6), (11)••Holds withhmnkl instead of bmnkl.Table 5
Characterizations ofΨ:ΛJt, where Ψ,Λ∈Mu,Cbp,Cp,Cr,Ls.
(Ψ↓:ΛJt→)J∞tJptJbptJrtJs1t1≤s1<∞Mu(5)(6), (8), (9)(5), (6), (17), (18), (19), (20)••Cp•(5), (6), (7), (8), (9)(5), (6), (7), (8), (9)(5), (6), (7), (8), (9)•Cbp(5)(5), (6), (7), (13), (14)(5), (6), (7), (13), (14)(5), (6), (7), (13), (14)•Cr•(5), (6), (7), (15), (16)(5), (6), (7), (15), (16)(5), (6), (7), (15), (16)•Ls0<s≤1(10)•(6), (10)•(12)Ls1<s<∞(11)•(6), (11)••Holds withumnkl instead of bmnkl.


## 5. Conclusions

As we have stated some of them in Table1, double sequence spaces which are obtained by using the domains of triangular matrices have been studied many times by many authors in the past. In the light of these and similar studies, in the current study, we described a new RH-regular and triangular 4D matrix derived by famous Jordan’s function which is an arithmetic function and introduced some double sequence spaces by using this matrix on some classical double sequence spaces. Moreover, we investigated their some properties and inclusion relations related them, computed duals, and characterized some matrix classes. We conclude that the results obtained from the 4D Jordan totient matrix Jt is more general and extensive than the existent results of the authors [6, 7]. We expect that our results might be a reference for further studies in this field.



---

*Source: 5594751-2021-05-19.xml*