ON SOME PROPERTIES OF CONTINUOUS G-FRAMES AND RIESZ-TYPE CONTINUOUS G-FRAMES
Mohammad Reza Abdollahpour and Yavar khedmati
Department of Mathematics, Faculty of Sciences,
University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran
e-mails: [email protected]; [email protected];
[email protected], [email protected]
(Received 6 October 2015; accepted 31 March 2016)
In this paper we provide some necessary and sufficient conditions under which, a family of bounded operators is a continuousg-frame (Riesz-type continuous g-frame). Also, we study stability of duals of continuousg-frames.
Key words : Continuous frame; continuousg-frame; Riesz-type continuousg-frame.
1. INTRODUCTION
In 1952, the concept of frames for Hilbert spaces was defined by Duffin and Schaeffer [4]. Frames
are important tools in the signal processing, image processing, data compression, etc. In 1993, Ali et
al., [2] developed the notion of ordinary frame to a family indexed by a measurable space which are
known as continuous frames.
Definition 1.1 — LetHbe a complex Hilbert space and(Ω, µ)be a measure space. A mapping
F : Ω→ His called a continuous frame with respect to(Ω, µ), if
(i) F is weakly-measurable, i.e., for allf ∈ H,ω→ hf, F(ω)iis a measurable function onΩ,
(ii) there exist constantsAF, BF >0such that
AFkfk2 ≤
Z
Ω
|hf, F(ω)i|2dµ(ω)≤BFkfk2, f ∈ H.
In 2006, g-frames or generalized frames introduced by Sun [6] and Abdollahpour and Faroughi
of this paper we assume thatHis a complex Hilbert space and(Ω, µ)is a measure space with positive
measureµand{Kω : ω ∈ Ω}is a family of Hilbert spaces. Now, we summarize some facts about
continuousg-frames from [1].
We say that F ∈ Qω∈ΩKω is strongly measurable if F as a mapping of Ω to
L
ω∈ΩKω is
measurable, where
Y
ω∈Ω Kω=
(
f : Ω→ [ ω∈Ω
Kω:f(ω)∈ Kω
)
.
Definition 1.2 — We say thatΛ = {Λω ∈ B(H,Kω) : ω ∈ Ω}is a continuousg-frame forH
with respect to{Kω:ω ∈Ω}if
(i) for eachf ∈ H,{Λωf :ω∈Ω}is strongly measurable,
(ii) there are two constants0< AΛ≤BΛ<∞such that
AΛkfk2 ≤
Z
Ω
kΛω(f)k2dµ(ω)≤BΛkfk2, f ∈ H. (1.1)
We call AΛ, BΛ the lower and upper continuous g-frame bounds, respectively. Λ is called a
tight continuousg-frame ifAΛ = BΛ,and a Parseval continuousg-frame ifAΛ = BΛ = 1.If for
eachω ∈ Ω,K = Kω,then Λ is called a continuousg-frame forH with respect to K. A family
Λ = {Λω ∈ B(H,Kω) : ω ∈ Ω} is called a continuous g-Bessel family for H with respect to {Kω :ω ∈Ω}if the right hand inequality in (1.1) holds for allf ∈ H. In this case,BΛis called the
Bessel constant.
If there is no confusion, we use continuousg-frame (continuousg-Bessel family) instead of
con-tinuousg-frame forHwith respect to{Kω:ω∈Ω}(continuousg-Bessel family forHwith respect
to{Kω :ω∈Ω}).
Proposition 1.3 [1] — LetΛ = {Λω ∈ B(H,Kω) : ω ∈ Ω} be a continuous g-frame. There
exists a unique positive and invertible operatorSΛ:H → Hsuch that for eachf, g∈ H
hSΛf, gi=
Z
Ω
hf,Λ∗ωΛωgidµ(ω), f, g∈ H,
andAΛI ≤SΛ ≤BΛI.
The operatorSΛin Proposition 1.3 is called the continuousg-frame operator ofΛ.Also, we have
hf, gi =
Z
Ω
hSΛ−1f,Λω∗Λωgidµ(ω)
=
Z
Ω
We consider the space
b
K =
(
F ∈ Y ω∈Ω
Kω :F is strongly measurable,
Z
Ω
kF(ω)k2dµ(ω)<∞
)
.
It is clear thatKbis a Hilbert space with point wise operations and the inner product given by
hF, Gi=
Z
Ω
hF(ω), G(ω)idµ(ω).
Proposition 1.4 [1] — LetΛ = {Λω ∈ B(H,Kω) : ω ∈ Ω}be a continuous g-Bessel family.
Then the mappingTΛ:K → Hb defined by
hTΛF, gi=
Z
Ω
hΛ∗ωF(ω), gidµ(ω), F ∈Kb, g∈ H, (1.3)
is linear and bounded withkTΛk ≤ √
BΛ.Also, for eachg∈ Hwe have
TΛ∗(g)(ω) = Λωg, ω∈Ω.
Theorem 1.5 [1] — Let(Ω, µ)be a measure space, whereµisσ-finite. Suppose thatΛ ={Λω ∈ B(H,Kω) : ω ∈Ω}is a family of operators such that{Λωf : ω ∈Ω}is strongly measurable, for
eachf ∈ H.ThenΛis a continuousg-frame if and only if the operatorTΛ:K → Hb defined by (1.3)
is a bounded and onto operator.
The operatorsTΛandTΛ∗in Theorem 1.8 are called synthesis and analysis operators ofΛ,
respec-tively.
Definition 1.6 — LetΛ ={Λω ∈B(H,Kω) :ω∈Ω}andΘ ={Θω ∈B(H,Kω) :ω ∈Ω}be
two continuousg-frames such that
hf, gi=
Z
Ω
hf,Θ∗ωΛωgidµ(ω), f, g∈ H,
thenΘis called a dual continuousg-frame ofΛ.
LetΛ ={Λω ∈B(H,Kω) :ω∈Ω}be a continuousg-frame. ThenΛ =e {ΛωSΛ−1 ∈B(H,Kω) : ω ∈ Ω}is a continuousg-frame and by (1.2),Λe is a dual of Λ.we callΛe the canonical dual ofΛ.
One can always get a tight continuousg-frame from any continuousg-frame, in fact, ifΛ ={Λω ∈ B(H,Kω) : ω ∈ Ω} is a continuous g-frame for Hthen {ΛωSΛ−1/2 ∈ B(H,Kω) : ω ∈ Ω} is a
Definition 1.7 — LetΛ ={Λω ∈B(H,Kω) :ω ∈Ω}andΘ ={Θω ∈B(K,Kω) :ω ∈Ω}be
two continuousg-frames. ThenΛandΘare said similar if there is an invertible operatorS:H → K
such thatΘωS = Λωfor a.e.ω ∈Ω.
Two continuousg-Bessel familiesΛ ={Λω∈B(H,Kω) :ω∈Ω}andΘ ={Θω∈B(H,Kω) : ω∈Ω}are weakly equal, if for allf ∈ H,
Λωf = Θωf, a.e. ω∈Ω.
Proposition 1.8 [1] — LetΛ ={Λω ∈B(H,Kω) :ω∈Ω}andΘ ={Θω∈B(K,Kω) :ω∈Ω}
be two continuousg-frames. ThenΛandΘare similar if and only if their analysis operators have the
same ranges (weakly).
If the continuous g-frameΛ = {Λω ∈ B(H,Kω) : ω ∈ Ω}have only one dual(weakly), i.e.,
every dual ofΛis weakly equal to the canonical dual ofΛ, thenΛis called a Riesz-type continuous
g-frame.
Theorem 1.9 — LetΛ = {Λω ∈ B(H,Kω) : ω ∈ Ω} be a continuousg-frame. ThenΛ is a
Riesz-type continuousg-frame if and only ifRangeTΛ∗ =Kb.
2. SOMENEWRESULTS FORCONTINUOUSG-FRAMES
In this section by generalizing some results of [5], we give some necessary and sufficient conditions
that a family of bounded operators{Λω ∈B(H,Kω) :ω ∈ Ω}is a continuousg-frame (Riesz-type
continuousg-frame). In this section until the end of Proposition 2.5,Kωis a finite dimensional Hilbert
space with orthonormal basis{eω,j :j∈Jω}for allω∈Ω.
Proposition 2.1 — Let{ei}i∈I be an orthonormal basis forHand{eωj}j∈Jω be an orthonormal
basis forKω, for allω∈Ω. Then the following are equivalent:
(i) Λ ={Λω ∈B(H,Kω) :ω∈Ω}is a Parseval continuousg-frame.
(ii) There exist an orthonormal set{ψi}i∈I ⊆ Kb such that
P
i∈Ikψi(ω)k2 < ∞for a.e. ω ∈ Ω
and for allf ∈ H,
Λωf =
X
i∈I
hf, eiiψi(ω), a.e. ω∈Ω.
PROOF:(i)⇒(ii)We considerψi=TΛ∗ei, for alli∈I. Then
kψik2 = kTΛ∗eik2 =
Z
Ω
and
hψi, ψji = hTΛ∗ei, TΛ∗eji=
Z
Ω
hΛωei,Λωejidµ(ω) =hei, eji= 0, i6=j.
On the other hand,
X
i∈I
kψi(ω)k2 =
X
i∈I
kΛωeik2 =
X
i∈I
kX
j∈Jω
hΛωei, eωjieωjk2
= X
j∈Jω
X
i∈I
|hei,Λ∗ωeωji|2=
X
j∈Jω
kΛ∗ωeωjk2 <∞.
Furthermore, for anyf ∈ H,
Λωf =
X
i∈I
hf, eiiΛωei =
X
i∈I
hf, eiiψi(ω).
Conversely, since{ψi}i∈I⊆Kbis an orthonormal set then for anyf ∈ H
Z
Ω
kΛωfk2dµ(ω) =
Z Ω ° ° °X i∈I
hf, eiiψi(ω)
° ° °2dµ(ω)
= kX i∈I
hf, eiiψik2
= X
i∈I
|hf, eii|2 =kfk2.
We recall that a Riesz basis for a Hilbert spaceHis a family of the form{U ei}i∈I, where{ei}i∈I
is an orthonormal basis forHandU :H → His a bounded bijective operator. 2
Theorem 2.2 — LetΛω∈B(H,Kω)be given for allω∈Ω.The following are equivalent:
(i) Λ ={Λω ∈B(H,Kω) :ω∈Ω}is a continuous g-frame.
(ii) Λωf =
P
i∈Ihf, eiiψi(ω) for some orthonormal basis {ei}i∈I of H and some family {ψi}i∈I ⊆ Kb with the property that {ψi}i∈I is a Riesz basis for span{ψi}i∈I and
P
i∈Ikψi(ω)k2 <∞for a.e.ω ∈Ω.
(iii) Λωf =
P
i∈Ihf, hiiψi(ω) for some Riesz basis {hi}i∈I of H and some orthonormal set {ψi}i∈I⊆Kband with the property that
P
i∈Ikψi(ω)k2 <∞for a.e.ω ∈Ω.
(iv) Λωf =
P
i∈Ihf, hiiψi(ω)for some Riesz basis{hi}i∈I ofHand some family{ψi}i∈I ⊆ Kb
with the property that{ψi}i∈Iis a Riesz basis forspan{ψi}i∈I and
P
i∈Ikψi(ω)k2 <∞for
PROOF:(i)⇒(ii)Let{ei}i∈Ibe an orthonormal basis ofHandψi =TΛ∗eifor alli∈I. Then
for any finite sequence of scalars{ck},we have
kXckψkk2=k
X
ckTΛ∗ekk2 = kTΛ∗
¡ X
ckek
¢
k2
=
Z
Ω kΛω
¡ X
ckek
¢
k2dµ(ω),
So
AΛX|ck|2 =AΛkXckekk2 ≤ kXckψkk2
≤ BΛkXckekk2 =BΛX|ck|2.
Therefore, by [3, Theorem 3.6.6],{ψi}i∈Iis a Riesz basis forspan{ψi}i∈I. We also haveΛωf
=Pi∈Ihf, eiiψi(ω)and
P
i∈Ikψi(ω)k2 <∞.
(ii)⇒(i)Since{ψi}i∈I ⊆Kbis a Riesz basis forspan{ψi}i∈I, there existB ≥A >0such that
Akfk2=AX i∈I
|hf, eii|2 ≤ k
X
i∈I
hf, eiiψik2
≤ BX
i∈I
|hf, eii|2 =Bkfk2.
Thus
Akfk2 ≤
Z
Ω
kΛωfk2dµ(ω) =kX i∈I
hf, eiiψik2 ≤Bkfk2.
(i)⇒(iii)Let{ei}i∈Ibe an orthonormal basis ofH. By Proposition 2.1, there exists an orthonormal
set{ψi}i∈IinKbsuch that
P
i∈Ikψi(ω)k2 <∞for a.e.ω ∈Ω, and for allg∈ H
ΛωSΛ−1/2g=
X
i∈I
hg, eiiψi(ω), a.e. ω∈Ω.
For anyf ∈ Hthere existg∈ Hsuch thatSΛ−1/2g=f. Thus
Λωf = ΛωSΛ−1/2g=
X
i∈I
hSΛ1/2f, eiiψi(ω) =
X
i∈I
hf, SΛ1/2eiiψi(ω).
It is sufficient to takehi =SΛ1/2ei, for alli∈I.
hi =U eifor alli∈I. Then for anyf ∈ Hwe have
Z
Ω
kΛωfk2dµ(ω) =
Z
Ω
kX
i∈I
hf, hiiψi(ω)k2dµ(ω)
= kX i∈I
hf, hiiψik2
= X
i∈I
|hf, hii|2=X
i∈I
|hU∗f, eii|2.
So
kU−1k−2kfk2 ≤
Z
Ω
kΛωfk2dµ(ω) =X
i∈I
|hf, hii|2 ≤ kUk2kfk2, f ∈ H.
(iv)⇒(i)There existC, D >0such that
Ckfk2 ≤X i∈I
|hf, hii|2≤Dkfk2, f ∈ H.
On the other hand, letB≥A >0be the bounds of Riesz basis{ψi}i∈I.Then
ACkfk2≤AX i∈I
|hf, hii|2 ≤ k
X
i∈I
hf, hiiψik2
=
Z
Ω
kX
i∈I
hf, hiiψi(ω)k2dµ(ω)
≤ BX
i∈I
|hf, hii|2 ≤BDkfk2.
for allf ∈ H.According to the(i)⇒(ii)the implication(i)⇒(iv)is obvious. 2
Proposition 2.3 — The following are equivalent:
(i) There exists a Riesz-type continuousg-frameΛ ={Λω ∈B(H,Kω) :ω∈Ω}forH.
(ii) There exist an orthonormal basis{ψi}i∈IforKbsuch that
X
i∈I
kψi(ω)k2 <∞, a.e. ω∈Ω.
(iii) For every orthonormal basis{φi}i∈IforKbwe have
X
i∈I
PROOF : (i) ⇒ (ii) By Theorem 2.2,Λωf =
P
i∈Ihf, hiiψi(ω) for some Riesz basis{hi}i∈I
ofH and some orthonormal set{ψi}i∈I ⊆ Kb with the property that
P
i∈Ikψi(ω)k2 < ∞ for a.e. ω∈Ω.By Theorem 1.9, for anyψinKbthere existsf ∈ Hsuch thatTΛ∗f =ψ.Thus
ψ=X
i∈I
hf, hiiψi.
So,span{ψi}i∈I=Kb, therefore{ψi}i∈Iis an orthonormal basis forKb.
(ii)⇒(i)Let us defineΛω ∈B(H,Kω)byΛωf =
P
i∈Ihf, eiiψi(ω)for allω∈Ω,where{ei}i∈I
is an orthonormal basis forH.By Proposition 2.1, Λ = {Λω ∈ B(H,Kω) : ω ∈ Ω}is a Parseval
continuousg-frame. For anyψinKbwe haveψ=Pi∈Ihψ, ψiiψi.So we have
TΛ∗³ X j∈I
hψ, ψjiej
´
(ω) = Λω
³ X
j∈I
hψ, ψjiej
´
= X
i∈I
X
j∈I
hψ, ψjiej, ei
®
ψi(ω)
= X
i∈I
hψ, ψiiψi(ω) =ψ(ω),
for allω ∈Ω.This means thatTΛis surjective and by Theorem 1.9, the proof is complete. (iii)⇒(ii)It is clear.
(i) ⇒ (iii) By assumption, Γ = {ΛωSΛ−1/2 : ω ∈ Ω} is a Parseval continuous g-frame and RangeT∗
Γ = RangeTΛ∗ = Kb. Therefore, TΓ∗ is a surjective isometry and so TΓ∗ is unitary. Let {φi}i∈Ibe an orthonormal basis forKb. Let us considerei =TΓφi, for alli∈I. Then,{ei}i∈I is an
orthonormal basis forHand
X
i∈I
kφi(ω)k2 =
X
i∈I
k X
j∈Jω
hφi(ω), eωjieωjk2
= X
i∈I
X
j∈Jω
|hφi(ω), eωji|2
= X
i∈I
X
j∈Jω
|hΛωSΛ−1/2ei, eωji|2
= X
j∈Jω
X
i∈I
|hei, SΛ−1/2Λ∗ωeωji|2
= X
j∈Jω
kSΛ−1/2Λ∗ωeωjk2 <∞,
Corollary 2.4 — LetM be a closed subspace ofKb. Then the following are equivalent:
(i) There exists a continuousg-frameΛ ={Λω∈B(H,Kω) :ω∈Ω}forHandRangeTΛ∗ =M.
(ii) There exist an orthonormal basis{ψi}i∈IforM with the property that
P
i∈Ikψi(ω)k2 <∞,
for a.e.ω∈Ω.
(iii) Every orthonormal basis{φi}i∈I forM satisfies
P
i∈Ikφi(ω)k2 <∞, for a.e. ω∈Ω.
PROOF: (i) ⇒ (ii) SinceΛ is a continuousg-frame,Γ = {ΛωSΛ−1/2 ∈ B(H,Kω) : ω ∈ Ω}
is a Parseval continuousg-frame. Let{ei}i∈Ibe an orthonormal basis forH.If we takeψi =TΓ∗ei,
for all i ∈ I then by Proposition 2.1, {ψi}i∈I is an orthonormal set of M with the property that
P
i∈Ikψi(ω)k2 <∞for a.e.ω∈Ωand
ΛωSΛ−1/2g=
X
i∈I
hg, eiiψi(ω), a.e. ω∈Ω, g∈ H.
For anyψ∈M there existsf ∈ Hsuch thatTΛ∗f =ψ.Letg=SΛ1/2f,then
ψ(ω) = Λωf = ΛωSΛ−1/2g=
X
i∈I
hg, eiiψi(ω).
(ii)⇒(i)Let{ei}i∈I be an orthonormal basis forH.We define bounded operator
Λω:H → Kω, Λωf =
X
i∈I
hf, eiiψi(ω),
for a.e. ω ∈Ω.Then by Proposition 2.1,Λ ={Λω ∈B(H,Kω) : ω ∈ Ω}is a parseval continuous g-frame. Ifψ∈M,thenψ=Pi∈Ihψ, ψiiψiand
TΛ∗³ X i∈I
hψ, ψiiei
´
(ω) =ψ(ω).
Thus, the proof is complete. (iii)⇒(ii)It is clear.
(i)⇒(iii)Let{φi}i∈Ibe an orthonormal basis forM. LetΓ ={ΛωSΛ−1/2 ∈B(H,Kω) :ω ∈Ω},
thenTΓTΓ∗ =TΓ∗TΓ =I.If we takeei=TΓφi,for alli∈Ithen{ei}i∈Iis an orthonormal basis for Hand
X
i∈I
kφi(ω)k2=
X
j∈jω
kSΛ−1/2Λ∗ωeωjk
Proposition 2.5 — Let Λ = {Λω ∈ B(H,Kω) : ω ∈ Ω} and Θ = {Θω ∈ B(H,Kω) : ω ∈ Ω} be two continuous g-frames with representation Λωf =
P
i∈Ihf, eiiψi(ω) and Θωf =
P
i∈Ihf, fiiφi(ω)for some{φi}i∈Iand{ψi}i∈IinKb,and for some orthonormal bases{ei}i∈I and {fi}i∈IforH. ThenΛis a dual ofΘif and only if
hψi, φji=hei, fji, i, j∈I.
In particular, ifei=fi, for alli∈I,thenΛis a dual ofΘif and only if{ψi}i∈I and{φi}i∈I are
biorthogonal.
PROOF: For anyf, g∈ Hwe have
hf, gi = X
i,j∈I
hf, eiihfj, gihei, fji,
and
Z
Ω
hf,Λ∗ωΘωgidµω =
Z
Ω
X
i,j∈I
hf, eiihfj, gihψi(ω), φj(ω)idµ(ω)
= X
i,j∈I
hf, eiihfj, gihψi, φji.
Thus, the proof is completed. 2
In the rest of this section we intend to generalize some results of [7] to continuousg-frame.
Lemma 2.6 — Let (Ω, µ) be a measure space. Let Λ = {Λω ∈ B(H,Kω) : ω ∈ Ω} and
Θ = {Θω ∈ B(H,Kω) : ω ∈ Ω} be two Parseval continuous g-frames. Then Range(TΘ∗) ⊆ Range(T∗
Λ)if and only if there exists an isometryU :H → Hsuch thatΘω = ΛωU for a.e.ω ∈Ω.
PROOF : Let Range(TΘ∗) ⊆ Range(TΛ∗). RangeTΛ∗ is a closed subspace of Kb. We take
P = T∗
ΛTΛ.So, Range(P) = Range(TΛ∗).Since Λ is a Parseval continuous g-frame, then P is
an orthogonal projection fromKbontoRange(T∗
Λ).We considerU =TΛTΘ∗. Then
kU fk2 =hU∗U f, fi=hTΘP TΘ∗f, fi=kfk2, f ∈ H.
and
hU∗f , gi = hTΘTΛ∗f , gi=
Z
Ω
Thus
Z
Ω
k(ΛωU −Θω)fk2dµ(ω) =
Z
Ω
kΛωU(f)k2dµ(ω)−
Z
Ω
hΛωU(f),Θωfidµ(ω)
−
Z
Ω
hΘωf ,ΛωU(f)idµ(ω) +
Z
Ω
kΘωfk2dµ(ω)
=kU fk2− hU∗U f , fi − hU f , U fi+kfk2
=kU fk2− kU fk2− kU fk2+kfk2= 0.
Hence,ΛωU f = Θωf for a.e.ω∈Ωandf ∈ H. The other implication is clear. 2
Proposition 2.7 — Let (Ω, µ) be a measure space. LetΛ = {Λω ∈ B(H,Kω) : ω ∈ Ω}and
Θ = {Θω ∈ B(H,Kω) : ω ∈ Ω} be two Parseval continuous g-frames and Λ be a Riesz-type
continuousg-frame. ThenΘis Riesz-type if and only if there exists an unitary operatorU :H → K
such thatΘωU = Λω for a.e.ω∈Ω.
PROOF: Let Θ be Riesz-type, thenRange(TΛ∗) = Range(TΘ∗) = Kb. By Lemma 2.6, there
is a bounded operatorU on H such that U∗U = IH andΘω = ΛωU for a.e. ω ∈ Ωand hence T∗
Θ =TΛ∗U. Since bothTΛ∗andTΘ∗ are invertible, it follows thatU is a unitary operator. For the other
implication, letU :H → Hbe a unitary linear operator such thatΘω = ΛωU, thenTΘ∗ =TΛ∗U and
so,RangeTΘ∗ =Kb. By Theorem 1.9,Θis Riesz-type. 2
Proposition 2.8 — Let Λ = {Λω ∈ B(H,Kω) : ω ∈ Ω} be a Riesz-type continuousg-frame
andΘ = {Θω ∈B(H,Kω) : ω ∈ Ω}be a continuousg-frame. ThenΘis a Riesz-type continuous g-frame if and only ifΛandΘare similar.
PROOF: It is obvious by Proposition 1.8 and Theorem 1.9. 2
Theorem 2.9 — LetΛ = {Λω ∈ B(H,Kω) : ω ∈ Ω}be a Riesz-type continuous g-frame and
Θ = {Θω ∈ B(H,Kω) : ω ∈ Ω} be a continuous g-frame. Then the following conditions are
equivalent:
(i) Θis a Riesz-type continuous g-frame.
(ii)There exists a constantM >0such that for allφ∈Kb
PROOF:(i)⇒(ii)By Proposition 2.8, there is an invertible bounded operatorU :H → Hsuch
thatΘω = ΛωU for a.e.ω ∈Ω.For anyg∈ Handφ∈Kbwe have
hU∗TΛφ, gi = hTΛφ, U gi=
Z
Ω
hU∗Λ∗ωφ(ω), gidµ(ω)
=
Z
Ω
hΘ∗ωφ(ω), gidµ(ω) =hTΘφ, gi.
ThenTΘ=U∗TΛand so
kTΛφ−TΘφk2 =kTΛφ−U∗TΛφk2 ≤ kI−U∗k2kTΛφk2
≤ (1 +kUk)2kTΛφk2.
Similarity
kTΛφ−TΘφk2 ≤ (1 +kU−1k)2kTΘφk2.
Then we have
kTΛφ−TΘφk2 ≤M.min{kTΛφk2,kTΘφk2}
whereM =max{(1 +kUk)2,(1 +kU−1k)2}.
(ii) ⇒ (i)Since TΛis invertible, then for any f ∈ H there exists an uniqueφ ∈ Kb such that TΛφ=f. Let us consider well defined bounded operatorU :H → Hby
U f =TΘφ, f ∈ H.
By the inequality (2.1), U is injective. On the other hand, since TΘ is surjective, then U is
surjective. So,U is invertible. For anyφ∈Kbandg∈ Hwe have
hφ,{(Θω−ΛωU∗)g}ω∈Ωi=
Z
Ω
hφ(ω),(Θω−ΛωU∗)gidµ(ω)
=
Z
Ω
hΘ∗ωφ(ω), gidµ(ω)−
Z
Ω
hΛ∗ωφ(ω), U∗gidµ(ω)
=hTΘφ, gi − hTΛφ, U∗gi
=hU TΛφ, gi − hU TΛφ, gi= 0
3. STABILITY OFDUALS OF CONTINUOUSG-FRAMES
In this section we study stability of duals of continuousg-frames.
Lemma 3.1 — Λ = {Λω ∈ B(H,Kω) : ω ∈ Ω}be a continuous g-frame and Θ = {Θω ∈ B(H,Kω) : ω ∈ Ω}is a dual ofΛ. Then there exists a operatorS ∈B(H,Kb)such thatTΛS = 0
andΘωf = (Sf)(ω) + ΛωSΛ−1f,for allω ∈Ωandf ∈ H.
PROOF: Let us considerS :H →Kb, by
(Sf)(ω) = Θωf −ΛωSΛ−1f, f ∈ H, ω∈Ω.
Then for allf ∈ H,
kSfk= (
Z
Ω
k(Sf)(ω)k2dµ(ω))1/2
≤(
Z
Ω
kΘωfk2dµ(ω))1/2+ (
Z
Ω
kΛωSΛ−1fk2dµ(ω))1/2
≤(pBΘ+√1
AΛ)kfk.
Thus,Sis bounded. On the other hand, for anyf, g∈ H,
hTΛSf , gi=
Z
Ω
hΛ∗ω(Sf)(ω), gidµ(ω)
=
Z
Ω
hΛ∗ωΘωf , gidµ(ω)−
Z
Ω
hΛ∗ωΛωSΛ−1f , gidµ(ω)
=hf, gi − hf, gi= 0.
So,TΛS = 0. 2
Theorem 3.2 — LetΛ ={Λω ∈B(H,Kω) :ω ∈Ω}andΘ ={Θω ∈B(H,Kω) : ω ∈Ω}be
two continuous g-frames. Also, letΛ =ˆ {Λˆω ∈B(H,Kω) :ω ∈Ω}be a fixed dual forΛ. IfΛ−Θ
is a continuous g-Bessel family with sufficiently small boundε >0, then there exists a dualΘˆ forΘ
such thatΛˆ−Θˆ is also continuous g-Bessel.
PROOF: By Lemma 3.1, there existS∈B(H,Kb)such thatTΛS= 0and
ˆ
Λωf = (Sf)(ω) + ΛωSΛ−1f, f ∈ H, ω ∈Ω.
LetM ={Mω∈B(H,Kω) :ω∈Ω}such that
It is easy to see thatM is a continuousg-Bessel family with bound(√1
AΘ +kSk)
2. Forf, g∈ Hwe
have
hf−TΘTM∗ f , gi=hf, gi −
Z
Ω
hΘ∗ω(TM∗ f)(ω), gidµ(ω)
=hf, gi −
Z
Ω
hΘ∗ωMωf , gidµ(ω)
=hf, gi −
Z
Ω
hΘ∗ω(Sf)(ω), gidµ(ω)−
Z
Ω
hΘ∗ωΘωSΘ−1f , gidµ(ω)
=hf, gi − hTΘSf , gi − hf, gi
=−hTΘSf , gi.
Thus,f−TΘTM∗ f =−TΘSf,for allf ∈ H.Hence
kf −TΘTM∗ fk=kTΘSfk=kTΘSf −TΛSfk
≤ kTΘ−TΛkkSkkfk
≤√εkSkkfk,
Therefore, for allf ∈ H,kIH −TΘTM∗ k ≤ √
εkSkand thus,TΘTM∗ is invertible becauseεis
sufficiently small. Hence,Θ =ˆ {Θˆω ∈ B(H,Kω) : ω ∈Ω},Θˆω = Mω(TΘTM∗ )−1 is a dual forΘ.
Because
Z
Ω
hΘˆωf ,Θωgidµ(ω) =
Z
Ω
D
(TM∗ (TΘTM∗ )−1f)(ω),Θωg
E
dµ(ω)
=
Z
Ω
hTΘTM∗ (TΘTM∗ )−1f , gidµ(ω)
=hf, gi.
And we have
kSΛ−SΘk=kTΛTΛ∗−TΛTΘ∗ +TΛTΘ∗ −TΘTΘ∗k
≤ kTΛ−TΘk(kTΛk+kTΘk)
≤√ε(pBΛ+
p
BΘ).
(3.1)
And byΛ(e f) = (Sf)(ω) + ΛωSΛ−1f andMω(f) = (Sf)(ω) + ΘωSΘ−1f, we have
hTΛˆφ−TMφ, fi=
Z
Ω
hφ(ω),Λˆωfidµ(ω)−
Z
Ω
hφ(ω), Mωfidµ(ω)
=
Z
Ω
hφ(ω),ΛωSΛ−1f−ΘωSΘ−1fidµ(ω)
=hTΛφ, SΛ−1fi − hTΘφ, SΘ−1fi
Hence, by inequality (3.1) we have
kTΛˆφ−TMφk
≤ kTΛφkkSΛ−1−S−Θ1k+kTΛ−TΘkkφkkSΘ−1k
≤ kTΛkkφkkSΛ−1kkSΘ−SΛkkSΘ−1k+A1 Θ
√ εkφk
≤ 1
AΛ
1
AΘ √
εpBΛ(
p
BΛ+
p
BΘ)kφk+A1 Θ
√ εkφk
= √
ε(AΛ+BΛ+ √
BΛBΘ) AΛAΘ kφk.
(3.2)
If we takeT = (TΘTM∗ )−1, then
kTk ≤ 1
1− kIH −T−1k ≤
1 1−√εS,
and so
kIH −Tk ≤ kTkkIH −T−1k ≤ √
εkSk
1−√εkSk. (3.3)
Consequently, by inequalities (3.2) and (3.3)
kTΛˆφ−TΘˆφk=supkfk=1|hTΛˆφ−TΘˆφ, fi|
=supkfk=1|h(IH−T∗)TΛˆφ+T∗(TΛˆ −TM)φ, fi|
≤ kIH −T∗kkTΛˆkkφk+kT∗kk(TΛˆ −TM)φk
≤ √
εkφk 1−√εkSk(kSk
q
BΛˆ +AΛ+BΛ+ √
BΛBΘ) AΛAΘ ).2
REFERENCES
1. M. R. Abdollahpour and M. H. Faroughi, Continuous G-Frames in Hilbert Spaces, Southeast Asian Bulletin of Mathematics, 32 (2008), 1-19.
2. S. T. Ali, J.-P. Antoine and J.-P. Gazeau, Continuous frames in Hilbert space, Annals of Physics, 222 (1993), 1-37.
4. R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Transactions of the American Mathematical Society, 72 (1952), 341-366.
5. J.-P. Gabardo and D. Han, Frames associated with measurable space, Advances in Computational Math-ematics, 18 (2003), 127-147.
6. W. Sun,G-frames andg-Riesz bases, Journal of Mathematical Analysis and Applications, 322 (2006), 437-452.
