CHARACTEIZATIONS OF WOVEN g-FRAMES AND WEAVING g-FRAMES IN HILBERT SPACES AND C∗-MODULES∗

Faculty of Mathematical Sciences and Computer, Kharazmi University,599 Taleghani Ave., Tehran 15618, Iran

E-mail: khosravi_amir@yahoo.com; khosravi@khu.ac.ir; mr.farmanis@gmail.com

Abstract In this paper,using Parseval frames we generalize Sun’s results to g-frames in Hilbert C∗-modules.Moreover,for g-frames in Hilbert spaces,we present some characterizations in terms of a family of frames,not only for orthonormal bases.Also,we have a note about a comment and a relation in the proof of Proposition 5.3 in [D.Li et al.,On weaving g-frames for Hilbert spaces,Complex Analysis and Operator Theory,2020].Finally,we have some results for g-Riesz bases,woven and P-woven g-frames.

Key words g-frame;fusion frame;woven frame;Riesz basis;g-Riesz basis

1 Introduction

Frames for Hilbert spaces were first introduced by Duffin and Schaeffer [1],to study nonharmonic Fourier series in 1952.Daubechies,Grossmann and Meyer (2017 Abel Prize winner)in their fundamental paper [2]reintroduced frames and from then on frame theory popularized(see [3,4]).

Sun in [5]introduced g-frames in Hilbert spaces and presented some characterizations for them.Later,the first author et al.in [6]introduced g-frames and fusion frames in HilbertC∗-modules.Recently,Bemrose,Casazza,Gröchenig,Lammers and Lynch in [7]introduced a new concept of weaving frames which is motivated by a problem regarding distributed signal processing.For a finite number of g-frames,the notions of woven and weaving (P-woven) are defined,which have some applications in distributed wireless sensor networks,too (see [8]).Many interesting and useful results of weaving frames and weaving g-frames are obtained,see[6,8–13].

In the rest of this section we state some definitions.In Section 2,we define woven and P-woven g-frames in HilbertC∗-modules and using Parseval frames we present some characterizations for them and generalize some results in [5]to g-frames in HilbertC∗-modules.We remark that all the results in Section 2 are valid for g-frames in Hilbert spaces.In Section 3,we present some characterizations for g-frames in Hilbert spaces with respect to frames (not necessarily Parseval frames) and we get some results for g-frames and g-Riesz bases and we generalize Theorem 3.1 in [14]to g-frames in Hilbert spaces.We also consider a claim in the proof of Proposition 5.3 in [13]and we improve their result.In Section 4,we get some results for woven and P-woven g-frames.

Throughout this paper except Section 2,H,K,HiandKi,wherei ∈I,denote separable Hilbert spaces andI,JandIiare finite or countable subsets of Z.Also,for everyi ∈I,B(H,Ki)is the set of all bounded linear operators fromHtoKi,andB(H,H) is denoted byB(H).

Definition 1.1A family of vectors{fi}i∈Iin a Hilbert spaceHis said to be a frame if there are constants 0

whereAandBare lower frame bound and upper frame bound,respectively.

A frame is called a tight frame ifA=B,and is called a Parseval frame ifA=B=1.If a sequence{fi}i∈Isatisfies the upper bound condition,then{fi}i∈Iis called a Bessel sequence.

Nowadays frame theory is a standard notion in applied mathematics,computer science,engineering,physics,probability,data processing and many other fields.However,technical advances and massive amount of data increased demands for generalizations of frames and so many generalizations of frames have been introduced,e.g.oblique frames [5],pseudo-frames[15],outer frames [16],fusion frames [12,14,17]and g-frames [5,14].

For each sequence{Ki:i ∈I} of Hilbert spaces,we define the space⊕i∈IKiby

which is a Hilbert space with the inner product defined by

Definition 1.2A sequence Λ={Λi ∈B(H,Ki):i ∈I} is called a generalized frame,or simply a g-frame,forHwith respect to{Ki:i ∈I} if there are two positive constantsAandBsuch that

We callAandBthe lower and upper frame bounds,respectively,and{Λi:i ∈I} is called a tight g-frame ifA=Band a Parseval g-frame ifA=B=1.

If only the right-hand side inequality is required,Λ is a g-Bessel sequence.

If Λ is a g-Bessel sequence,then the synthesis operator for Λ is the linear operator,

The adjoint of synthesis operator is called the analysis operator.The analysis operator is the linear operator,

Definition 1.3A sequence{Λi ∈B(H,Ki):i ∈I} is called

(1) g-complete,if{f:Λif=0,i ∈I}={0},

(2) a g-Riesz basis forHwith respect to{Ki}i∈I,if{Λi ∈B(H,Ki):i ∈I} is g-complete and there exist two positive constantsAandBsuch that for each finite subsetJ ⊆Iandgj ∈Ki,

(3)a near g-Riesz basis,if there exists a finite subsetσofIfor which{Λi}i∈Iσis a g-Riesz basis forHwith respect to{Ki}i∈Iσ.

Definition 1.4A sequence{Λi ∈B(H,Ki):i ∈I} is called a g-Riesz sequence if it is a g-Riesz basis for

Definition 1.5Let{Wi:i ∈I} be a sequence of closed subspaces ofH,{ωi:i ∈I} ⊆ℓ∞(I) such thatωi>0,for eachi ∈I.The sequenceW={(Wi,ωi) :i ∈I} is said to be a fusion frame forH,if there exist constants 0

whereπWiis the orthogonal projection ontoWi.The constantsAandBare called fusion frame bounds.A fusion frameW={(Wi,ωi):i ∈I} is called a tight fusion frame if the constantsAandBcan be chosen so thatA=B.IfA=B=1 we say that it is a Parseval fusion frame.If only the right hand side inequality is required,it is called a Bessel fusion sequence.

2 g-frames in Hilbert C∗-modules

In this section we generalize Sun’s results to HilbertC∗-modules,and since every orthonormal basis is a Parseval frame and Parseval frame is more suitable for HilbertC∗-modules,using Parseval frames we present some characterizations for g-frames.First we recall some definitions.

In this section,letHandKi,for eachi ∈I,be finitely or countably generated HilbertC∗-modules overC∗-algebraAand letB(H,Ki) denote the set of all adjointable operators fromHtoKi.We note that frames,g-frames,g-Bessel sequences and fusion frames are defined as in Hilbert spaces except that norm is replaced byA-valued norm and everyWiis a closed orthogonally complemented submodule ofH(see [18]).

Definition 2.1Let Λ={Λi ∈B(H,Ki):i ∈I}.Then Λ is called a g-frame inHwith respect to{Ki:i ∈I} if there exist constantsA,B>0 such that

Theorem 2.2Let Λ={Λi ∈B(H,Ki) :i ∈I} and for eachi ∈I,{fij:j ∈Ii} be a Parseval frame forKi.Then

(i) Λ={Λi:i ∈I} is a g-Bessel sequence if and only ifϕ={(fij):i ∈I,j ∈Ii} is a Bessel sequence and in this case their frame operators are the same,SΛ=Sϕ.

(ii) Λ is a g-frame if and only ifϕis a frame.

ProofLetx ∈H.Thenand since for eachi ∈I,Λix ∈Kiand{fij:j ∈Ii} is a Parseval frame forKi,then

HenceSΛ=Sϕ,which yields the results.

Corollary 2.3LetW={(Wi,vi) :i ∈I},whereWiis a closed orthogonally complemented submodule ofHandvibe a positive constant for eachi ∈I.Let also{fj:j ∈J}be a Parseval frame forH.Then{(Wi,vi):i ∈I} is a fusion frame if and only ifϕ={viπWi(fj):i ∈I,j ∈J} is a frame forHand their frame operators are the same,Sϕ=SW.

ProofPlainly,Wis a fusion frame if and only if{viπWi:i ∈I} is a g-frame and their frame operators are the same.Now since{fj:j ∈J} is a Parseval frame forH,then for eachi ∈I,{πWi(fj) :j ∈J} is a Parseval frame forWiand by the above theorem we have the result.

Definition 2.4Letϕ={ϕi:i ∈I} andψ={ψi:i ∈I} be Bessel sequences inH.ThenSϕ,ψ:H →His defined byfor everyx ∈H.Then{ψi:i ∈I}is called a dual frame of{ϕi:i ∈I} ifSϕ,ψ=IH,whereIHis the identity operator onH.Alsoψis called an approximate dual of{ϕi:i ∈I} if‖Sϕ,ψ-IH‖<1.

Definition 2.5Let Λ={Λi ∈B(H,Ki) :i ∈I} and Γ={Γi ∈B(H,Ki) :i ∈I} be g-Bessel sequences.ThenSΛ,Γ:H →His defined by

IfSΛ,Γ=IH,then{Γi:i ∈I} is a g-dual of{Λi:i ∈I} and if‖SΛ,Γ-IH‖<1,then{Γi:i ∈I} is an approximate g-dual of{Λi:i ∈I} (see [19]).

Theorem 2.6Let Λ={Λi ∈B(H,Ki) :i ∈I},Γ={Γi ∈B(H,Ki) :i ∈I} and for eachi ∈I,{fij:j ∈Ii} be a Parseval frame forKi.Then

(i) Γ is a g-dual of Λ if and only ifψ={(fij) :i ∈I,j ∈Ii} is a dual frame ofϕ={(fij):i ∈I,j ∈Ii} andSΛ,Γ=Sϕ,ψ.

(ii) Γ is an approximate g-dual of Λ if and only ifψis an approximate dual ofϕ.

ProofBy Theorem 2.2,Γ={Γi:i ∈I} is a g-Bessel sequence if and only ifψis a Bessel sequence.Similar result holds for Λ andϕ.Also for everyx ∈H,and for eachi ∈I,Λix ∈Kiand{fij:j ∈Ii} is a Parseval frame forKi,then

Therefore (i) and (ii) follow.Also (iii) follows from Theorem 2.2 and (i).

Weaving frames were introduced in [9]and weaving g-frames were introduced by Li et al.in [13]and they have potential applications in wireless sensor networks.In the sequel of this section,we introduce a P-woven family of g-Bessel sequences and get some results for woven g-frames and P-woven g-Bessel sequences (see [8]).

Definition 2.7A family{∈B(H,Ki) :i ∈I} forj=1,2,···,m,of g-frames forHis said to be woven if there existA,B>0 such that for each partitionP={σ1,σ2,···,σm} ofI,its corresponding weaving ΛP={∈B(H,Ki):i ∈σj,j=1,2,···,m} is a g-frame forHwith respect to{Ki:i ∈I} with lower and upper boundsAandB,respectively.

Definition 2.8A family{∈B(H,Ki) :i ∈I} forj=1,2,···,m,of g-Bessel sequences forHis said to beP-woven (weaving) if there exists a partitionP={σ1,σ2,···,σm}ofIsuch that its corresponding weaving is a g-frame forHwith respect to{Ki:i ∈I}.

Proposition 2.9Let{∈B(H,Ki):i ∈I},forj=1,2,···,m,be a family of g-frames and for eachi ∈I,{fij:j ∈Ii} be a Parseval frame forKi.Then

(i){∈B(H,Ki):i ∈I}forj=1,2,···,m,are woven g-frames forHif and only if there existA,B>0 such that for every partitionP′={δ1,δ2,···,δm} ofJ={(i,ϑ):i ∈I,ϑ ∈Ii},whereδj={(i,ϑ):i ∈σj,ϑ ∈Ii} andP={σ1,σ2,···,σm} is a partition ofIand

(ii) If{()∗(fiϑ) :i ∈I,ϑ ∈Ii},j=1,2,···,m,are woven frames,then{:i ∈I} forj=1,2,···,mare woven g-frames.

Proof(i)We know that{∈B(H,Ki):i ∈I},forj=1,2,···,m,are woven g-frames if and only if there existA,B>0 such that for each partitionP={σ1,σ2,···,σm} ofI,

Now by Theorem 2.2,this is equivalent to

and we have the result.

(ii) It follows from (i).

Remark 2.10The proof of (i) shows that if{:i ∈I},j=1,···,m,is a P-woven family of g-Bessel sequences,then the family of Bessel sequences{()∗(fiv) : (i,ν)∈J},j=1,···,m,is P-woven.

3 g-frames in Hilbert Spaces

In this section we give some characterizations for g-frames and we get some results for g-Riesz bases.

Theorem 3.1Let{Λi ∈B(H,Ki) :i ∈I} and{fij:j ∈Ii} be a frame forKiwith bounds,0

ProofLetf ∈H.Then Λif ∈Kifor eachi ∈Iand so

Consequently,

Now if{Λi ∈B(H,Ki):i ∈I}is a g-frame with boundsC,D,then by(3.2),{(Λi)∗(fij):i ∈I,j ∈Ii} is a frame with boundsACandBD.

Conversely,if{(Λi)∗(fij) :i ∈I,j ∈Ii} is a frame with boundsC′,D′,then by (3.2),{Λi ∈B(H,Ki):i ∈I} is a g-frame with bounds

In this note,using Parseval frames we present a characterization for g-frames,which is very useful for fusion frames.

Theorem 3.2Let{Λi ∈B(H,Ki):i ∈I} and for eachi ∈I,{fij:j ∈Ii} be a Parseval frame forKi.Then

(i){Λi:i ∈I} is a g-frame inHwith respect to{Ki:i ∈I} (g-Bessel sequence) if and only if{(Λi)∗(fij):i ∈I,j ∈Ii} is a frame inH(Bessel sequence).

(ii) If{Λi:i ∈I} is a g-frame,then≥dimH,and the equality holds whenever{Λi:i ∈I} is a g-Riesz basis.

Proof(i) SinceA=Ai=Bi=B=1,by Theorem 3.1,{Λi:i ∈I} is a g-frame with boundsCandDif and only if{(fij):i ∈I,j ∈Ii} is a frame forHwith boundsCandD.

(ii) If for eachi ∈Iwe take{fij:j ∈Ii} an orthonormal basis,by (i) and the fact that in every space the cardinal of each frame is greater than or equal to the dimension of the space we have≥dimH.

If{Λi:i ∈I} is a g-Riesz basis,then{(fij):i ∈I,j ∈Ii} is a Riesz basis forHand therefore the dimension ofHis equal to the cardinality of{(fij) :i ∈I,j ∈Ii} which is equal todim(Ki),where|Ii| denotes the cardinality ofIi,for eachi ∈Iand we have the result,see [5].

Proposition 3.3Let{Λi ∈B(H,Ki) :i ∈I},K=⊕i∈IKiand{fj:j ∈J} be a Parseval frame forK.Then{Λi:i ∈I} is a g-frame forHwith respect to{Ki:i ∈I} if and only if{πKi(fj):i ∈I,j ∈J}is a frame forH.Moreover the g-frame operator of{Λi:i ∈I}and frame operator of{πKi(fj):i ∈I,j ∈J} are the same,where for eachi ∈I,πKiis the orthogonal projection ontoKi.

ProofBy consideringK=⊕i∈IKi,we can take eachKias a closed subspace ofKand if we take{fj:j ∈J}a Parseval frame forK,then for eachi ∈I,{πKifj:j ∈J}is a Parseval frame forKiand we have the result.

Corollary 3.4Let{fj:j ∈J} be a Parseval frame forH.Then{(Wi,υi) :i ∈I} is a fusion frame if and only if{υi(πWi(fj)):i ∈I,j ∈J} is a frame forH.

ProofIf{fj:j ∈J} is a Parseval frame forH,then for eachi ∈I,{πWi(fj) :j ∈J}is a Parseval frame forWi.Then{(Wi,υi):i ∈I} is a fusion frame if and only if{υiπWi(fj):i ∈I,j ∈J} is a frame forH(see [20]).

Theorem 3.5Let{Λi ∈B(H,Ki):i ∈I} and{fij:j ∈Ii} be a Riesz basis ofKiwith boundsAi,Bisuch thatThen{Λi:i ∈I} is a g-Riesz basis if and only if{(fij):i ∈I,j ∈Ii} is a Riesz basis.

ProofWe note that{Λi ∈B(H,Ki):i ∈I}is g-complete if and only if{f ∈H:Λif=0 for eachi ∈I}={0}.Also Λi(f)=0,for eachi ∈I,if and only if〈f,(fij)〉=0,for eachi ∈I,j ∈Ii.Hence{Λi ∈B(H,Ki):i ∈I} is g-complete if and only if{(fij):i ∈I,j ∈Ii}is complete.Since{fij:j ∈Ii} is a Riesz basis with boundsAi,Bi,then for eachgi ∈Kiwe havefor some complex numberscij,and

Now if{Λi ∈B(H,Ki):i ∈I}is a g-Riesz basis with boundsC,D,then for each(cij)(i,j)∈J′∈ℓ2(J′),whereJ′={(i,j):i ∈I,j ∈Ii},we have

and therefore

Consequently,from (3.3) it follows that

Conversely,let{(fij) :i ∈I,j ∈Ii} be a Riesz basis with boundsC′,D′.For eachgi ∈Kiwe haveand by (1),

and we have the result.

Theorem 3.6Let{Λi ∈B(H,Ki):i ∈I}and for eachi ∈I,{fij:j ∈Ii}be a Parseval frame forKi.If{(Λi)∗(fij):i ∈I,j ∈Ii} is a Riesz basis,then{Λi:i ∈I} is a g-Riesz basis.Conversely,if{Λi:i ∈I} is a g-Riesz basis and for eachi ∈I,there existsmi>0 such that for every complex numbers{cij:i ∈I,j ∈Ii},

then{(fij):i ∈I,j ∈Ii} is a Riesz basis.

Since this relation holds for allcij,then〈fiν,fij〉=0 ifν≠jand〈fiν,fij〉=1 ifν=j,i.e.,{fiν:ν ∈Ii} is an orthonormal basis forKiand by Sun’s theorem we have the result.

Remark 3.7We note that in the proof of [13,Proposition 5.3]the authors claimed that for a g-frame{Λi ∈B(H,Ki) :i ∈I} ifTiis invertible for eachi ∈I,then automatically{ΛiTi ∈B(H,Ki):i ∈I} is a g-frame.Also they used the following inequality for g-frames,

in which it is not clear thatiin the right hand side is arbitrary or existing.But in the following example we show that both of them are not true in general.

Example 3.8LetHbe a Hilbert space andT ∈B(H) be invertible.For every natural number n,letThen{Λn ∈B(H):n ∈N} is a g-frame with boundsFor eachn ∈N,by takingTn=(Λn)-1=nT-1,eachTnis invertible and{ΛnTn ∈B(H):n ∈N}={IH:n ∈N} is not a g-frame,because for each non-zerox ∈H,

Also letx ∈Hbe non-zero.Then there exists a natural numberNsuch that for everyn ≥N,‖Tx-x‖>‖x‖/2 and therefore

Consequently,inequality (3.4) does not hold in general.

Remark 3.9We note that in the proof of Proposition 5.3 in [13],the first inequality is not valid in general.If we consider,we get a contradiction sinceA/B ≤1.

Now we state a result.

Proposition 3.10Let{Λi ∈B(H,Ki) :i ∈I} be a g-frame forHwith respect to{Ki:i ∈I} and (αi)i∈Ibe a sequence of complex numbers.

(i) IfTi=αiT,whereT ∈B(H) is invertible and there existm,M>0 such thatm ≤|αi|≤M,for eachi ∈I,then{ΛiTi:i ∈I}={αiΛiT:i ∈I} is a g-frame.

(ii) IfTi=αiIH,then{Λi:i ∈I} and{ΛiTi:i ∈I} are woven.

ProofLet{Λi:i ∈I} be a g-frame with boundsAandB.For everyx ∈H,we haveHence

and so (i) holds.

(ii) For everyσ ⊆Iand everyx ∈Hwe have

and similarly

which completes the proof.

4 Weaving g-frames

In the next theorem we try to find some relations between the operators corresponding to a weaving g-frame.

Theorem 4.1Let{∈B(H,Ki) :i ∈I} be a g-Bessel sequence forHwith Bessel boundBjand frame operatorSj,for eachj=1,2,···,m.Then

(ii){:i ∈I} forj=1,2,···,mare woven if and only if for each partitionP={σ1,σ2,···,σm} ofI,there existsAP>0 such thatAP·I ≤SP.

Proof(i) Plainly,and since for every partitionP={σ1,σ2,···,σm} ofIand everyf ∈H,

we have the result.

(ii) By Theorem 3.7 in [12]we have the result.

The next theorem gives sufficient condition for a finite number of g-Bessel sequences to be P-woven.

Theorem 4.2Let{∈B(H,Ki) :i ∈I} forj=1,2,···,m,be a family of g-Bessel sequences forHwith respect to{Ki:i ∈I}.Suppose that there exists a partitionP={σ1,σ2,···,σm} ofIsuch that

Then the family{:i ∈I},j=1,2,···,m,is P-woven.

Sincehi ∈Kiis arbitrary,we get that

LetAjandBjbe the lower and upper bounds of{Λji ∈B(H,Ki) :i ∈σj},respectively,for eachj=1,2,···,m.Then

Hence the family{∈B(H,Ki),i ∈I},forj=1,···,m,is P-woven.

Lemma 4.3Let{Λi ∈B(H,Ki) :i ∈I} be a g-frame and{Γi ∈B(H,Ki) :i ∈I} be a sequence for which there exist 0<λ1,λ2<1 such that for everyi ∈Iand eachx ∈Hwe have

Then{Λi ∈B(H,Ki):i ∈I} and{Γi ∈B(H,Ki):i ∈I} are woven g-frames.

ProofIf{Λi ∈B(H,Ki):i ∈I} is a g-frame with boundsA,B,then for everyi ∈I

So for eachx ∈Hwe have

Therefore{Λi ∈B(H,Ki) :i ∈σ} ∪{Γi ∈B(H,Ki) :i ∈σc} is a g-frame with boundsConsequently,{Λi:i ∈I} and{Γi:i ∈I} are woven g-frames.

Proposition 4.4LetH,K,HiandKibe Hilbert spaces for eachi ∈Iand let{∈B(H,Hi) :i ∈I} forj=1,2,···,mbe a woven family of g-frames.LetT ∈B(K,H)be invertible andTj i ∈B(Hi,Ki),for eachi ∈I,andj=1,···,msuch that for some 0<δ ≤M<∞we haveδ‖x‖ ≤‖(x)‖ ≤M‖x‖for everyx ∈Hi,i ∈Iandj=1,···,m.Then,j=1,···,mis a woven family of g-frames.

ProofBy the assumption there exist 0

Proposition 4.5Suppose{Λji ∈B(H,Ki) :i ∈I},j=1,2,···,m,of g-frames are woven with universal boundsAandB.IfF:⊕Ki →⊕Miis a bounded invertible operator such thatF(Ki)⊆Mi,for eachi,then{∈B(H,Mi) :i ∈I},j=1,2,···,mare also woven with universal boundsA‖F-1‖-2,B‖F‖2.

ProofIt is a known fact that if a g-frame has boundsAandB,then applying an invertible operatorFto it gives a g-frame with boundsA‖F-1‖-2andB‖F‖2.Let{∈B(H,Ki) :i ∈I},j=1,2,···,mbe a family of woven g-frames with universal boundsAandB.Then for each partitionP={σ1,σ2,···,σm} ofI,{∈B(H,Ki) :i ∈σj,j=1,2,···,m} is a g-frame with boundsAandB.Hence{∈B(H,Mi):i ∈σj,j=1,2,···,m} is a g-frame with boundsA‖F-1‖-2andB‖F‖2.Hence we have the result.

Now we state a result which is useful for Gabor frames and wavelets.

Proposition 4.6Let{∈B(H,Ki):i ∈I},forj=1,2,···,m,be a P-woven g-frame and for a partitionP={σ1,···,σm} ofI,ΛP={Λji ∈B(H,Ki) :i ∈σj,j=1,···,m} be a g-frame with boundsAandB,and also let for eachi ∈I,{∈B(Ki,Wi,ϑ) :ϑ ∈Ii} forℓ=1,2,···,ni,be aP-wowen g-frame forKiand for a partitionPi={δi,1,···,δi,ni} ofIi,(Γi)Pi={∈B(Ki,Wi,ν) :ν ∈δi,l,l=1,···,ni} be a g-frame with boundsCi,Disuch that 0

ProofBy the assumption,there exists a partitionP={σ1,σ2,···,σm} ofIsuch that{∈B(H,Ki) :i ∈σj,j=1,2,···,m} is a g-frame with boundsAandB,and for eachi ∈Ithere is a partitionPi={δi,1,δi,2,···,δi,ni} ofIisuch that{∈B(Ki,Wi,ϑ):ϑ ∈δi,l,l=1,···,ni} is a g-frame with boundsCi,Di.Now similarly to the proof of Proposition 4.4 we get that:i ∈σj,j=1,···,m;ϑ ∈δi,l,l=1,···,ni} is a g-frame with boundsAC,BD,which is the weaving g-frame corresponding to the partitionP′={δ1,δ2,···,δm} ofJ′={(i,ϑ) :i ∈I,ϑ ∈Ii},whereδj={(i,ϑ) :i ∈σj,ϑ ∈δi,ℓ,ℓ=1,···,ni}.Then we have

and similarly

and we have the result.

Conflict of InterestThe authors declare no conflict of interest.