The object of this paper is to introduce some new sequence spaces
related with the concept of lacunary strong almost convergence for double sequences and also to characterize these spaces through sublinear functionals that
both dominate and generate Banach limits and to establish some inclusion
relations.

1. Introduction and Preliminaries

Let w2 be the set of all real or complex double sequences. We mean the convergence in the Pringsheim sense, that is, a double sequence x=(xi,j)i,j=0∞ has a Pringsheim limit λ(denoted by P-limx=λ) provided that given ɛ>0 and there exists N∈ℕ such that |xi,j-λ|<ɛ whenever i,j≥N [1]. We denote by c2, the space of P-convergent sequences. A double sequence x=(xi,j) is bounded if ∥x∥=supi,j≥0|xi,j|<∞. Let l2∞ and c2∞ be the set of all real or complex bounded double sequences and the set of bounded and convergent double sequences, respectively. Moricz and Rhoades [2] defined the almost convergence of double sequences that x=(xi,j) is said to be almost convergent to a number λ if
(1.1)limp,q→∞supm,n≥0|1(p+1)(q+1)∑i=mm+p∑j=nn+qxi,j-λ|=0,
that is, the average value of (xi,j) taken over any rectangle
(1.2)D={(i,j):m≤i≤m+p,n≤j≤n+q},
tends to λ as both p and q tend to ∞ and this convergence is uniform in m and n. We denote the space of almost convergent double sequences by f2, as
(1.3)f2={x=(xi,j):limk,l→∞|tklpq(x)-λ|=0,uniformlyinp,q},
where
(1.4)tklpq(x)=1(k+1)(l+1)∑i=pp+k∑j=qq+lxi,j.

The notion of almost convergence for single sequences was introduced by Lorentz [3] and for double sequences by Moricz and Rhoades [2] and some further studies are in [4–14].

A double sequence x is called strongly almost convergent to a number λ if
(1.5)limk,l→∞1(k+1)(l+1)∑i=pp+k∑j=qq+l|xi,j-λe|=0,uniformlyinp,q.

By [f2], we denote the space of all strongly almost convergent double sequences. It is easy to see that the inclusions c2∞⊂[f2]⊂f2⊂l2∞ strictly hold. As in the case of single sequences, every almost convergent double sequence is bounded. But a convergent double sequence need not be bounded. Thus, a convergent double sequence need not be almost convergent. However every bounded convergent double sequence is almost convergent.

The notion of strong almost convergence for single sequences has been introduced by Maddox [15, 16] and for double sequences by Başarir [17].

A linear functional L on l2∞ is said to be Banach limit if it has the following properties [7],

L(x)≥0 if x≥0 (i.e., xi,j≥0 for all i, j),

L(e)=1, where e=(ei,j) with ei,j=1 for all i, j and

L(x)=L(S10x)=L(S01x)=L(S11x) where the shift operators S10x,S01x,S11x are defined by S10x=(xi+1,j),S01x=(xi,j+1),S11x=(xi+1,j+1).

Let B2 be the set of all Banach limits on l2∞. A double sequence x=(xi,j) is said to be almost convergent to a number λ if L(x)=λ for all L∈B2. If φ is any sublinear functional on l2∞, then we write {l2∞,φ} to denote the set of all linear functionals F on l2∞, such that F≤φ, that is, F(x)≤φ(x),∀x∈l2∞. A sublinear functional φ is said to generate Banach limits if F∈{l2∞,φ} implies that F is a Banach limit; φ is said to dominate Banach limits if F is a Banach limit implies that F∈{l2∞,φ}. Then if φ both generates and dominates Banach limits, then {l2∞,φ} is the set of all Banach limits.

Using the notations for single sequences, we present the notations for double-lacunary sequences that can be seen in [10]. The double sequence θr,s={(kr,ls)} is called a double-lacunary if there exist two increasing sequences of nonnegative integers such that k0=0,hr=kr-kr-1→∞ as r→∞ and l0=0,hs=ls-ls-1→∞ as s→∞. Let kr,s=krls,hr,s=hrhs,θr,s is determined by Ir,s={(i,j):kr-1<i≤krandls-1<j≤ls}. Also h-r,s=krls-kr-1ls-1 and θ=θ-r,s is determined by I-r,s={(i,j):kr-1<i≤kr∪ls-1<j≤ls}/(I1∪I2), where I1={(i,j):kr-1<i≤kr and ls<j<∞.} and I2={(i,j):ls-1<j≤ls and kr<i<∞.} with qr=kr/kr-1,qs=ls/ls-1 and qr,s=qrqs.

Das and Mishra [18] introduced the space of lacunary almost convergent sequences by combining the space of lacunary convergent sequences and the space of almost convergent sequences. Savaş and Patterson [10] extended the notions of lacunary almost convergence and lacunary strongly almost convergence to double-lacunary P-convergence and double-lacunary strongly almost P-convergence. They also established multidimensional analogues of Das and Patel's results.

We will use the following definition which may be called convergence in Pringsheim's sense with a bound:
(1.6)(xi,j-L)=O(1),(i,j⟶∞),
and also we will use the following definition which may be called convergence in Pringsheim's sense as follows:
(1.7)(xi,j-L)=o(1),(i,j⟶∞).

The following sequence spaces were introduced and examined by Başarir [19]:
(1.8)wθ={x:limrsupi1hr∑k∈Irtki(x-s)=0,forsomes},[w]θ={x:limrsupi1hr∑k∈Ir|tki(x-s)|=0,forsomes},[w1]θ={x:limrsupi1hr∑k∈Irtki(|x-s|)=0,forsomes},
with respect to sublinear functionals on l∞ (the set of all real or complex bounded single sequences) by
(1.9)ϕθ(x)=limr¯supi1hr∑k∈Irtki(x),ψθ(x)=limr¯supi1hr∑k∈Ir|tki(x)|,ζθ(x)=limr¯supi1hr∑k∈Irtki(|x|),
where tki(x)=(1/k)∑j=ii+k-1xj and |x|=(|xj|)j=1∞.

It can be easily seen that each of the above functionals are finite, well defined, and sublinear on l∞. There is a very close connection among these sequence spaces with the sublinear functionals which were given by Başarir [19]. Recently Mursaleen and Mohiuddine [7] generalized the sequence spaces which were studied by Das and Sahoo [20] for single sequences, to the double sequences as follows:
(1.10)w2={x=(xi,j):1(m+1)(n+1)∑k=0m∑l=0ntklpq(x-λe)⟶0,asm,n⟶∞,uniformlyinp,q,forsome∫∫∫},[w2]={x=(xi,j):1(m+1)(n+1)∑k=0m∑l=0n|tklpq(x-λe)|⟶0,asm,n⟶∞,uniformlyinp,q,forsomeλ∫∫∫},[w]2={x=(xi,j):1(m+1)(n+1)∑k=0m∑l=0ntklpq(|x-λe|)⟶0,asm,n⟶∞,uniformlyinp,q,forsomeλ∫∫∫}
by using (1.4).

The object of the present paper is to determine some new sublinear functionals involving double-lacunary sequence that both dominates and generates Banach limits. We also extend the sequence spaces which were introduced for single sequences by Başarir [19] to the double sequences with respect to these sublinear functionals. Furthermore, we present some inclusion relations with these new sequence spaces between the sequence spaces which were introduced by Mursaleen and Mohiuddine [7], earlier.

2. Sublinear Functionals and Double-Lacunary Sequence Spaces

In this section, we introduce the following sequence spaces:
(2.1)wθ2={x=(xi,j):P-limr,s→∞supp,q1h¯r,s∑(k,l)∈I¯r,stklpq(x-λe)=0,forsomeλ},[wθ2]={x=(xi,j):P-limr,s→∞supp,q1h¯r,s∑(k,l)∈I¯r,s|tklpq(x-λe)|=0,forsomeλ},[w]θ2={x=(xi,j):P-limr,s→∞supp,q1h¯r,s∑(k,l)∈I¯r,stklpq(|x-λe|)=0,forsomeλ},W(θ,2)={x=(xi,j):P-limr,s→∞1h¯r,s∑(k,l)∈I¯r,stkl00(x-λe)=0,forsomeλ},[W(θ,2)]={x=(xi,j):P-limr,s→∞supp,q1hr,s∑(k,l)∈I¯r,s|tkl00(x-λe)|=0,forsomeλ},W[θ,2]={x=(xi,j):P-limr,s→∞1hr,s∑(k,l)∈I¯r,stkl00(|x-λe|)=0,forsomeλ}.

It may be noted that almost convergent double sequences are necessarily bounded but the sequence spaces wθ2 and [wθ2] may contain unbounded sequences. Now we define the following functionals on l2∞ for a double-lacunary sequence θ=(θ-r,s) by,
(2.2)ϕθ2(x)=limr,s→∞¯supp,q1h¯r,s∑(k,l)∈I¯r,stklpq(x),ψθ2(x)=limr,s→∞¯supp,q1h¯r,s∑(k,l)∈I¯r,s|tklpq(x)|,φθ2(x)=limr,s→∞¯supp,q1h¯r,s∑(k,l)∈I¯r,stklpq(|x|),ζ2(x)=limk,l→∞¯supp,qtklpq(x),η2(x)=limk,l→∞¯supp,qtklpq(|x|).
It is easy to see that each of the above functionals are finite, well defined, and sublinear on l2∞.

Throughout the paper we will write limr,s for P-limr,s→∞ and by this notation we shall mean the convergence in the Pringsheim sense. In the following theorem, we demonstrate that {l2∞,ϕθ2} is the set of all Banach limits on l2∞ and characterize the space wθ2∩l2∞ in terms of the sublinear functional ϕθ2.

Theorem 2.1.

One has the following.

(1) The sublinear functional ϕθ2 both dominates and generates Banach limits, that is, ϕθ2(x)=ζ2(x), for all x=(xi,j)∈l2∞.

(1) From the definition of ζ2, for given ɛ>0 there exist k0,l0 such that
(2.4)tklpq(x)<ζ2(x)+ɛ,
for k≥k0,l≥l0 and for all p,q. This implies that
(2.5)ϕθ2(x)<ζ2(x)+ɛ,
for all x=(xi,j)∈l2∞. Since ɛ is arbitrary, so that ϕθ2(x)≤ζ2(x), for all x=(xi,j)∈l2∞ and hence
(2.6){l2∞,ϕθ2}⊂{l2∞,ζ2}=B2,
that is, ϕθ2 generates Banach limits.

Conversely, suppose that L∈B2. As L is the shift invariant, that is, L(S11x)=L(x)=L(S10x)=L(S01x) and using the properties of L∈B2, we obtain
(2.7)L(x)=L(1(k+1)(l+1)∑i=pp+k∑j=qq+lxi,j)=L(tklpq(x))=1h¯r,sL(∑(k,l)∈I¯r,stklpq(x))≤supp,q1h¯r,s∑(k,l)∈I¯r,stklpq(x).

It follows from the definition of ϕθ2, that for given ɛ>0 there exist r0,s0 such that
(2.8)1h¯r,s∑(k,l)∈I¯r,stklpq(x)<ϕθ2(x)+ɛ,
for r≥r0,s≥s0 and for all p,q. Hence by (2.8) and properties (1) and (2) of Banach limits, we have
(2.9)L(1h¯r,s∑(k,l)∈I¯r,stklpq(x))<L((ϕθ2(x)+ɛ)e)=ϕθ2(x)+ɛ,
for r≥r0,s≥s0 and for all p,q; where e=(ei,j) with ei,j=1 for all i, j. Since ɛ is arbitrary, it follows from (2.7) and (2.9) that
(2.10)L(x)≤ϕθ2(x),∀x=(xi,j)∈l2∞.
Hence
(2.11)B2⊂{l2∞,ϕθ2}.
That is, ϕθ2 dominates Banach limits. Combining (2.6) and (2.11), we get
(2.12){l2∞,ζ2}={l2∞,ϕθ2},
this implies that ϕθ2 dominates and generates Banach limits and ϕθ2(x)=ζ2(x) for all x∈l2∞.

(2) As a consequence of Hahn-Banach theorem, {l2∞,ϕθ2} is non empty and a linear functional F∈{l2∞,ϕθ2} is not necessarily uniquely defined at any particular value of x. This is evident in the manner the linear functionals are constructed. But in order that all the functionals {l2∞,ϕθ2} coincide at x=(xi,j), it is necessary and sufficient that
(2.13)ϕθ2(x)=-ϕθ2(-x),
we have
(2.14)limsupr,ssupp,q1h¯r,s∑(k,l)∈I¯r,stklpq(x)=liminfr,sinfp,q1h¯r,s∑(k,l)∈I¯r,stklpq(x).
But (2.14) holds if and only if
(2.15)1h¯r,s∑(k,l)∈I¯r,stklpq(x)⟶λ,asr,s⟶∞,uniformlyinp,q.
Hence, x=(xi,j)∈wθ2∩l2∞. But (2.13) is equivalent to ζ2(x)=-ζ2(-x), this holds if and only if x=(xi,j)∈f2. This completes the proof of the theorem.

If F(x-λe)=0 for all F∈{l2∞,ψθ2}, then we say that x=(xi,j) is ψθ2-convergent to λ. Similarly we define the φθ2-convergent sequences. In the following theorem we characterize the spaces [wθ2]∩l2∞ and [w]θ2∩l2∞ in terms of the sublinear functionals.

Theorem 2.2.

One has the following:

[wθ2]∩l2∞ = {x=(xi,j):ψθ2(x-λe)=0, for some λ} = {x=(xi,j):F(x-λe)=0, for all F∈{l2∞,ψθ2}, for some λ}

[w]θ2∩l2∞ = {x=(xi,j):φθ2(x-λe)=0, for some λ} = {x=(xi,j):F(x-λe)=0, for all F∈{l2∞,φθ2}, for some λ}.

Proof.

(1) It can be easily verified that x=(xi,j)∈[wθ2]∩l2∞ if and only if
(2.16)ψθ2(x-λe)=-ψθ2(λe-x).
Since ψθ2(x)=-ψθ2(-x) then (2.16) reduces to
(2.17)ψθ2(x-λe)=0.
Now if F∈{l2∞,ψθ2} then from (2.17) and linearity of F, we have
(2.18)F(x-λe)=0.
Conversely, suppose that F(x-λe)=0 for all F∈{l2∞,ψθ2} and hence by Hahn-Banach theorem, there exists F0∈{l2∞,ψθ2} such that F0(x)=ψθ2(x). Hence
(2.19)0=F0(x-λe)=ψθ2(x-λe).

(2) The proof is similar to the proof of (1), above.

3. Inclusion Relations

We establish here some inclusion relations between the sequence spaces defined in Section 2.

Theorem 3.1.

We have the following proper inclusions and the limit is preserved in each case.

[f2]⊂[w]θ2⊂[wθ2]⊂wθ2⊂W(θ,2).

[w]θ2⊂[wθ2]⊂[W(θ,2)]⊂W(θ,2).

[w]θ2⊂W[θ,2]⊂[W(θ,2)]⊂W(θ,2).

Proof.

(1) Let x∈[f2] with [f2]-limx=λ. Then
(3.1)tklpq(|x-λe|)⟶0,ask,l⟶∞,uniformlyinp,q.
This implies that
(3.2)1h¯r,s∑(k,l)∈I¯r,stklpq(|x-λe|)⟶0,ask,l⟶∞,uniformlyinp,q.
This proves that x∈[w]θ2 and [f2]-limx=[w]θ2-limx=λ. Since
(3.3)1h¯r,s|∑(k,l)∈I¯r,stklpq(x-λe)|≤1h¯r,s∑(k,l)∈I¯r,s|tklpq(x-λe)|≤1h¯r,s∑(k,l)∈I¯r,stklpq(|x-λe|),
this implies that [w]θ2⊂[wθ2]⊂wθ2 and [w]θ2-limx=[wθ2]-limx=wθ2-limx=λ. Since
(3.4)1h¯r,s∑(k,l)∈I¯r,stklpq(x-λe)
converges uniformly in p,q as r,s→∞, implies the convergence for p=0=q. It follows that wθ2⊂W(θ,2) and wθ2-limx=W(θ,2)-limx=λ. This completes the proof of (1).

It is easy to see the proof of (2) and (3). So we omit them.

Theorem 3.2.

One has the following proper inclusions;
(3.5)[f2]⊂([w]θ2∩l2∞)⊂([wθ2]∩l2∞)⊂f2.

Proof.

The proof of the theorem is similar as in [7, Theorem 4.2]. So we omit it.

Prior to giving Lemmas 3.3 and 3.5, we need the following notations used in [10]:
(3.6)IC1={(i,j):q≤j≤q+n,p+m<i<∞},IC2={(i,j):q+n<j<∞,p≤i≤p+m},Cp,qm,n={(i,j):p≤i≤p+morq≤j≤q+n}∖(IC1∪IC2),ID1={(i,j):q+(y+1)hs-1<j<∞,p+xhr≤i≤p+(x+1)hr-1},ID2={(i,j):q+yhs≤j≤q+(y+1)hs-1,p+(x+1)hr-1<i<∞},Dp,qx,y={(i,j):p+xhr≤i≤p+(x+1)hr-1orq+yhs≤j≤q+(y+1)hs-1}∖(ID1∪ID2).

Lemma 3.3.

Suppose ɛ>0 there exist m0,n0,p0, and q0 such that
(3.7)1|C0,0m,n|∑(k,l)∈C0,0m,n(1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l|xi,j-λe|)<ɛ,
for m≥m0,n≥n0 and p≥p0,q≥q0. Then x∈[w]2.

Proof.

Let ɛ>0 be given. Choose m01,n01,p0 and q0 such that
(3.8)1|C0,0m,n|∑(k,l)∈C0,0m,n(1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l|xi,j-λe|)<ɛ6
for m≥m01,n≥n01,p≥p0,q≥q0. We need only to show that given ɛ>0 there exist m02 and n02 such that
(3.9)1|C0,0m,n|∑(k,l)∈C0,0m,n(1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l|xi,j-λe|)<ɛ,
for m≥m02,n≥n02 and 0≤p≤p0,0≤q≤q0. If we take m0=max{m01,m02} and n0=max{n01,n02}, then (3.9) holds for m≥m0,n≥n0 and for all p and q, which gives the result. Once p0 and q0 have been chosen, they are fixed, so
(3.10)∑(k,l)∈C0,0p0,q0(1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp0,q0|xi,j-λe|)=M,
is finite. Now taking 0≤p≤p0, 0≤q≤q0 and m≥p0, n≥q0 we have from (3.8) and (3.10)
(3.11)1|C0,0m,n|∑(k,l)∈C0,0m,n(1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l|xi,j-λe|)=1|C0,0m,n|∑(k,l)∈C0,0p0,q0(1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp0,q0|xi,j-λe|)+1|C0,0m,n|∑(k,l)∈C0,0p0,q0(1|Cp,qp+k,q+l|∑(i,j)∈Cp0+1,q0+1p+k,q+l|xi,j-λe|)+1|C0,0m,n|∑(k,l)∈Cp0+1,q0+1m,n(1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp0,q0|xi,j-λe|)+1|C0,0m,n|∑(i,j)∈Cp0+1,q0+1m,n(1|Cp,qp+k,q+l|∑(i,j)∈Cp0+1,q0+1p+k,q+l|xi,j-λe|)≤M|C0,0m,n|+3⋅ɛ6=M|C0,0m,n|+ɛ2.
Therefore taking m and n sufficiently large, we can make M/|C0,0m,n|+ɛ/2<ɛ which gives (3.9) and hence the result.

Theorem 3.4.

We have [w]θ2=[w]2 for every θ¯r,s.

Proof.

Let x∈[w]θ2; then given ɛ>0 there exist r0,s0 and λ such that
(3.12)1h¯r,s∑(k,l)∈C0,0hr-1,hs-1tklpq(|x-λe|)<ɛ,
for r≥r0,s≥s0, p=kr-1+1+α and q=ls-1+1+β where α,β≥0. Let m≥hr such that m=δ1hr+θ1-1 where δ1 is an integer. Also let n≥hs such that n=δ2hs+θ2-1 where δ2 is an integer and 1≤θ1≤hr, 1≤θ2≤hs. Since m≥hr for δ1≥1 and n≥hs for δ2≥1 we have,
(3.13)1|C0,0m,n|∑(k,l)∈C0,0m,ntklpq(|x-λe|)≤1|C0,0m,n|∑(k,l)∈C0,0(δ1+1)hr-1,(δ2+1)hs-1tklpq(|x-λe|)=1|C0,0m,n|∑x,y=0,0δ1,δ2∑(k,l)∈D0,0x,ytklpq(|x-λe|)≤1|C0,0m,n|∑x,y=0,0δ1,δ2(h¯r,s⋅ɛ)≤(δ1+1)(δ2+1)|C0,0m,n|(h¯r,s⋅ɛ)=o(1),
which gives the result. Therefore by Lemma 3.3, [w]θ2⊂[w]2. It is clear that [w]2⊂[w]θ2 for every θ¯r,s. This completes the proof.

Lemma 3.5.

Suppose, for a given ɛ>0, there exist m0,n0,p0, and q0 such that
(3.14)1|C0,0m,n|∑(k,l)∈C0,0m,n|1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l(xi,j-λe)|<ɛ,
for all m≥m0,n≥n0 and p≥p0,q≥q0. Then x∈[w2].

Proof.

Let ɛ>0 be given and choose m0,n0,p0 and q0 such that
(3.15)1|C0,0m,n|∑(k,l)∈C0,0m,n|1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l(xi,j-λe)|<ɛ4
for all m≥m0,n≥n0,p≥p0, and q≥q0. As in Lemma 3.3, it is enough to show that there exist m01 and n01 such that for m≥m01,n≥n01 implies
(3.16)1|C0,0m,n|∑(k,l)∈C0,0m,n|1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l(xi,j-λe)|<ɛ,
for all p and q with 0≤p≤p0 and 0≤q≤q0. Since p0 and q0 are fixed,
(3.17)∑(k,l)∈C0,0p0,q0(1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp0,q0|xi,j-λe|)=M.
Now, let 0≤p≤p0,0≤q≤q0, and m≥p0,n≥q0 and consider the following;
(3.18)1|C0,0m,n|∑(k,l)∈C0,0m,n|1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l(xi,j-λe)|≤1|C0,0m,n|∑(k,l)∈C0,0p0,q0(1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp0,q0|xi,j-λe|)+1|C0,0m,n|∑(k,l)∈C0,0p0,q0|1|Cp,qp+k,q+l|∑(i,j)∈Cp0+1,q0+1p+k,q+l(xi,j-λe)|+1|C0,0m,n|∑(k,l)∈Cp0+1,q0+1m,n|1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l(xi,j-λe)|≤M|C0,0m,n|+1|C0,0m,n|∑(k,l)∈C0,0p0,q0|1|Cp,qp+k,q+l|∑(i,j)∈Cp0+1,q0+1p+k,q+l(xi,j-λe)|+1|C0,0m,n|∑(k,l)∈Cp0+1,q0+1m,n|tklpq(x-λe)|.
Let k-p0≥m01, then k+p-p0≥m01 for 0≤p≤p0. Also if we let l-q0≥n01, then l+q-q0≥n01 for 0≤q≤q0. Therefore from (3.15)
(3.19)1|C0,0m,n|∑(k,l)∈C0,0p0,q0|1|Cp,qp+k,q+l|∑(i,j)∈Cp0+1,q0+1p+k,q+l(xi,j-λe)|<1|C0,0p0,q0|∑(k,l)∈C0,0p0,q0|1|Cp0+1,q0+1p0+(p+k-p0),q0+(q+l-q0)|∑(i,j)∈Cp0+1,q0+1p0+(p+k-p0),q0+(q+l-q0)(xi,j-λe)|<ɛ4.
From (3.18) and (3.19)
(3.20)1|C0,0m,n|∑(k,l)∈C0,0m,n|1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l(xi,j-λe)|≤M|C0,0m,n|+2ɛ4<ɛ,
for sufficiently large values of m and n. Hence the result.

Theorem 3.6.

For every θ¯r,s, One has [wθ2]∩l2∞=[w2]∩l2∞.

Proof.

Let x∈[wθ2]∩l2∞. For ɛ>0, there exist r0,s0,p0 and q0 such that
(3.21)1h¯r,s∑(k,l)∈C0,0hr-1,hs-1|tklpq(x-λe)|<ɛ2
for r≥r0,s≥s0,p≥p0 and q≥q0 with p=kr-1+1+α where α≥0,q=ls-1+1+β and β≥0. Let m≥hr and n≥hs where m≥1 and n≥1. Then
(3.22)1|C0,0m,n|∑(k,l)∈C0,0m,n|1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l(xi,j-λe)|=1|C0,0m,n|∑(k,l)∈C0,0δ1hr-1,δ2hs-1|tklpq(x-λe)|+1|C0,0m,n|∑(k,l)∈Cδ1hr,δ2hsm,n|tklpq(x-λe)|≤1|C0,0m,n|∑x,y=0,0δ1-1,δ2-1∑(k,l)∈D0,0x,y|tklpq(x-λe)|+1|C0,0m,n|∑(k,l)∈Cδ1hr,δ2hsm,n(1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l|xi,j-λe|).
Since (xi,j)∈l2∞ for all i and j, there exists M such that |xi,j-λe|≤M. From (3.21) and (3.22), we have the following:
(3.23)1|C0,0m,n|∑(k,l)∈C0,0m,n|1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l(xi,j-λe)|≤δ1δ2|C0,0m,n|(h¯r,sɛ2)+Mh¯r,s|C0,0m,n|.
Thus for m and n sufficiently large, we have the following:
(3.24)1|C0,0m,n|∑(k,l)∈C0,0m,n|1|Cp,qp+k,q+l|∑(i,j)∈Cp,qp+k,q+l(xi,j-λe)|<ɛ,
for r≥r0,s≥s0 and p≥p0,q≥q0. Thus by Lemma 3.5, we have [wθ2]∩l2∞⊂[w2]∩l2∞. It is clear that [w2]∩l2∞⊂[wθ2]∩l2∞. This completes the proof of the theorem.

Corollary 3.7.

[f2]⊂[w]θ2⊂([wθ2]∩l2∞)⊂(wθ2∩l2∞)=f2.

Proof.

It is easy to see by combining Theorem 3.4, Theorem 3.6 with [7, Theorems 4.1 and 3.1(ii)]. So we omit it.

A paranormed space (X,g) is a topological linear space with the topology given by the paranorm g. It may be recalled that a paranorm g is a real subadditive function on X such that g(θ)=0,g(x)=g(-x) and scalar multiplication is continuous, that is, μn→μ,xn→x imply that (μnxn)→(μx) where μn,μ are scalars and xn,x∈X.

Let u=(uk,l) be a bounded double sequence of positive real numbers, that is, uk,l>0 for all k,l with supk,luk,l=H<∞. Let
(3.25)[wθ2(u)]={x=(xi,j):limr,ssupp,q1h¯r,s∑(k,l)∈I¯r,s|tklpq(x-λe)|uk,l=0forsomeλ}.
If u=(uk,l) is constant we write [wθ2]u in place of [wθ2(u)]. If we take u=(uk,l) with uk,l=1 for all k and l, then [wθ2(u)] is reduced to [wθ2] which is defined in Section 2.

Theorem 3.8.

Let u=(uk,l) be a bounded sequence of positive real numbers with supk,luk,l=H<∞. Then [wθ2(u)] is a complete linear topological space paranormed by
(3.26)g(x)=supr,s,p,q(1h¯r,s∑(k,l)∈I¯r,s|tklpq(x)|uk,l)1/M,
where M=max(1,H). In the case u is constant, [wθ2]u is a Banach space if u≥1 and is a p-normed space if 0<u<1.

Proof.

It is easy to see that [wθ2(u)] is a linear space with coordinatewise addition and scalar multiplication. Clearly g(θ)=0,g(x)=g(-x) and g is subadditive. To prove the continuity of multiplication, assume that x∈[wθ2(u)]. Since u=(uk,l) is bounded and positive there exists a constant δ>0 such that uk,l≥δ for all k,l. Now for |μ|≤1, |μ|uk,l≤|μ|δ and hence g(μx)≤|μ|δ/Mg(x). This proves the fact that g is a paranorm on [wθ2(u)].

To prove that [wθ2(u)] is complete, assume that (xm) is a Cauchy sequence in [wθ2(u)], that is, g(xm-xn)→0 as m,n→∞. Since
(3.27)1h¯r,s∑(k,l)∈I¯r,s|tklpq(xm-xn)|uk,l≤[g(xm-xn)]M,
it follows that |tklpq(xm-xn)|uk,l=0(1) as m,n→∞ for each k,l,p, and q. In particular
(3.28)t00pq(xm-xn)=|xp,qm-xp,qn|⟶0asm,n⟶∞,foreachfixedpandq.
Hence, (xm) is a Cauchy sequence in ℝ(or ℂ). Since ℝ(or ℂ) is complete, there exists x∈ℝ(or ℂ) such that xm→x coordinate wise as m→∞. It follows from (3.27) that given ɛ>0, there exists m0∈ℕ such that
(3.29)(1h¯r,s∑(k,l)∈I¯r,s|tklpq(xm-xn)|uk,l)1/M<ɛ,
for m, n>m0. Now making n→∞ and then taking supremum with respect to p and q in (3.29) we obtain g(xm-x)≤ɛ for m>m0. This proves that xm→x and x∈[wθ2(u)]. Hence [wθ2(u)] is complete. When u is constant, it is easy to derive the rest of the theorem.

Theorem 3.9.

Let 0<ρk,l≤σk,l<∞ for each k and l. Then [wθ2(ρ)]⊂[wθ2(σ)].

Proof.

Let x∈[wθ2(ρ)]. By the definition of [wθ2(ρ)], that for given ɛ>0 there exist r0,s0 such that
(3.30)1h¯r,s∑(k,l)∈I¯r,s|tklpq(x-λe)|ρk,l<ɛ,
for r>r0,s>s0 and for all p,q. Since 1/h¯r,s→0 as r,s→∞, then
(3.31)∑(k,l)∈I¯r,s|tklpq(x-λe)|ρk,l<∞,
for r>r0,s>s0 and for all p,q. This implies that
(3.32)|tklpq(x-λe)|<1,
for sufficiently large values of k,l and for all p,q. Then we get,
(3.33)1h¯r,s∑(k,l)∈I¯r,s|tklpq(x-λe)|σk,l≤1h¯r,s∑(k,l)∈I¯r,s|tklpq(x-λe)|ρk,l=o(1),
as r,s→∞ and for all p,q. Hence,
(3.34)limr,ssupp,q1h¯r,s∑(k,l)∈I¯r,s|tklpq(x-λe)|ρk,l=0,
is obtained and consequently we have x∈[wθ2(ρ)]. This completes the proof.

Theorem 3.10.

One has the following.

Let 0<infk,luk,l≤uk,l≤1 for each k and l. Then [wθ2(u)]⊂[wθ2].

Let 1≤uk,l≤supk,luk,l<∞ for each k and l. Then [wθ2]⊂[wθ2(u)].

Proof.

(1) It is clear from the above theorem. If we take ρk,l=uk,l and σk,l=1 for each k and l, then we have [wθ2(u)]⊂[wθ2].

(2) From the above theorem, if we take ρk,l=1 and σk,l=uk,l for each k and l, then we have [wθ2]⊂[wθ2(u)].

This completes the proof.

Acknowledgment

The authors are grateful to anonymous referees for their careful reading of the paper which improved it greatly.

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