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,0ProofLetf ∈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 0Proposition 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 0ProofBy 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.
