# New Properties of Dual ContinuousK-g-Frames in Hilbert Spaces

Authors: Abdeslam Touri; Hatim Labrigui; Mohamed Rossafi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

The concept of frames in Hilbert spaces continues to play a very interesting role in many kinds of applications. In this paper, we study the notion of dual continuousK-g-frames in Hilbert spaces. Also, we establish some new properties.


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## Body

## 1. Introduction and Preliminaries

The concept of frames in Hilbert spaces was introduced by Duffin and Schaeffer in 1952 [1] to study some deep problems in nonharmonic Fourier series; after the fundamental paper [2] by Daubechies, Grossman, and Meyer, frames’ theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames [3].LetX be a Banach space, Ω,μ a measure space, and function f:Ω⟶X a measurable function. The integral of the Banach-valued function f was defined by Bochner and others. Most properties of this integral are similar to those of the integral of real-valued functions.LetΩ,μ be a measure space, let U and V be two separable Hilbert spaces, Vww∈Ω is a sequence of subspaces of V, we denote ℬU,V as the collection of all bounded linear operators from U to V, and ℬU,U is abbreviated to ℬU.ForT∈ℬU,V, we use the notation ℛT and NT to denote, respectively, the range and the null space of T.Definition 1 (see [4]).
A sequenceΛω∈ℬU,Vωω∈Ω is called a continuous g-frame for U with respect to Vωω∈Ω if there exists 0<A<B<∞ such that(1)Ax2≤∫ΩΛωx2dμω≤Bx2,∀x∈U.
The numbersA and B are called lower and upper bounds of the continuous g-frame, respectively.
IfA=B=λ, the frame is λ-tight.
IfA=B=1, it is called a normalized tight continuous g-frame or a Parseval continuous g-frame.
We callΛω∈ℬU,Vωω∈Ω a continuous g-Bessel sequence if the right-hand inequality of (1) holds.LetΛ=Λw∈Ω be a continuous g-frame for U with respect to Vωω∈Ω.The synthesis operator ofΛ is defined by TΛ:⊕ω∈ΩVw⟶U, given by(2)TΛyww∈Ω=∫ΩΛω∗yωdμw,∀yww∈Ω∈⊕ω∈ΩVw,where ⊕ω∈ΩVw=yww∈Ω,yω∈Vω;∫Ωyω2dμω<∞.The existence of∫ΩΛω∗yωdμw implies that TΛ is well defined and bounded.The adjoint operator ofTΛ which is given by(3)TΛ∗:U⟶⊕ω∈ΩVw;TΛ∗x=Λωxw∈Ω,∀x∈U,is said to be the analysis operator of Λwω∈Ω.Definition 2 (see [5]).
LetK∈ℬU. A sequence Λω∈ℬU,Vωω∈Ω is called a continuous K-g-frame for U with respect to Vωω∈Ω if there exists 0<A<B<∞ such that(4)AK∗x2≤∫ΩΛωx2dμω≤Bx2,∀x∈U.
The numbersA and B are called lower and upper bounds of the continuous K-g-frame, respectively.
IfAK∗x2=∫ΩΛωx2dμω, the sequence Λωxw∈Ω is called a continuous K-g-frame for U with respect to Vωω∈Ω.
TheA-tight continuous K-g-frame for U is said to be a Parseval continuous K-g-frame if A=1.
Suppose thatK∈ℬU and Λ=Λω∈ℬU,Vωω∈Ω is a Parseval continuous K-g-frame for U with respect to Vωω∈Ω with synthesis operator TΛ. Then, it is easy to check(5)KK∗x=TΛTΛx=∫ΩΛω∗Λωxdμω,∀x∈U.Definition 3.
LetK∈ℬU and Λωω∈Ω∈ℬU,Vω be a continuous K-g-frame for U with respect to Vωω∈Ω. A continuous K-g-Bessel sequence Γωω∈Ω for U with respect to Vωω∈Ω is said to be a dual continuous K-g-Bessel sequence of Λωω∈Ω if(6)Kx=∫ΩΛω∗Γωxdμw,∀x∈U.Remark 1.
(1)
IfK=IdU, then a dual continuous K-g-Bessel sequence is just a dual continuous g-frame(2)
A dual continuousK-g-Bessel sequence is necessarily a continuous K∗-g-frame(3)
Denote the synthesis operator ofΛωω∈Ω and Γωω∈Ω by TΛ and TΓ, respectively; then, (6) means that K=TΛTΓ∗Lemma 1 (see [6]).
Suppose thatT∈ℬU,V has a closed range; then, there exists a unique operator T+∈ℬU,V, called the pseudo-inverse of T, satisfying(7)TT+T=T,T+TT+=T+,TT+∗=TT+,T+T∗=T+T,T∗+=T+∗NT+=ℛT⊥,ℛT+=NT⊥.Lemma 2 (see [7]).
LetU, U1, and U2 be three Hilbert spaces; also, let T∈ℬU1,U and K∈ℬU2,U. The following statements are equivalent:(1)
ℛT⊂ℛK(2)
There existsλ>0 such that TT∗≤λKK∗(3)
There existsθ∈ℬU1,U2 such that T=Kθ
Moreover, if (1)–(3) are valid, then there exists a unique operatorθ such that(a)
θ2=infμ:TT∗≤μKK∗(b)
NT=Nθ(c)
ℛθ⊂ℛK∗¯In the following section, we set out to generalize some results of Xiang [8]; in other words, we characterize the concept of dual continuous K-g-frames in Hilbert spaces, and we give some properties. Our results extend, generalize, and improve many existing results.


## 2. Main Result

Proposition 1.
Suppose thatK∈ℬU has a closed range, and let Λ=Λω∈ℬU,Vωω∈Ω to be a continuous K-g-frame for U with respect to Vωω∈Ω and Γ=Γωω∈Ω be a dual continuous K-g-Bessel sequence of Λωω∈Ω; then, ΓωK∗ω∈Ω is a continuous K-g-frame for U with respect to Vωω∈Ω.Proof.
For eachx∈U, we have Kx=∫ΩΛω∗Γωxdμw, and then(8)Kx2=supy=1Kx,y2=supy=1∫ΩΓωx,Λωydμω2≤supy=1∫ΩΓωx2dμω∫ΩΛωy2dμω≤BΛ∫ΩΓωx2dμω,which give(9)BΛ−1Kx2≤∫ΩΓωx2dμω≤BΓx2,where BΛ and BΓ are the upper bounds of Λ and Γ, respectively.
Consequently,(10)BΛ−1KK∗x2≤∫ΩΓωK∗x2dμω≤BΓK∗x2≤BΓK2x2.
Assume thatK∗ has a closed range; hence, for each y∈ℛK∗, we get y=K∗K∗+y=K+Ky, by Lemma 1, and thus,(11)y2=K+Ky2≤K+2Ky2.
Now,(12)BΛ−1K+−2K∗x2≤BΛ−1KK∗x2≤∫ΩΓωK∗x2dμω≤BΓK2x2,which gives ΓωK∗ω∈Ω is a continuous K-g-frame for U with respect to Vωω∈Ω with bounds BΛ−1K+−2 and BΓK2.Proposition 2.
LetK∈ℬU; then, every continuous K-g-frame admits a dual continuous K-g-Bessel sequence.Proof.
LetΛ=Λω∈ℬU,Vωω∈Ω be a continuous K-g-frame for U with respect to Vωω∈Ω, with bounds CΛ and DΛ; then, for each x∈U, we have(13)CΛK∗x,K∗x=CΛK∗x2≤∫ΩΛωx2dμω=TΛ∗x2=TΛ∗x,TΛ∗x,which is equivalent to CΛKK∗≤TΛTΛ∗; by Lemma 2, there exists θ∈ℬU,⊕ω∈ΩVw such that K=TΛθ.
LetPω be the projection on ⊕ω∈ΩVw that maps each element to its w-th component, i.e., Pωyzz∈Ω=uzz∈Ω, where uz=yω if z=ω, and uz=0 if not.
If we letΓω=Pωθ, then(14)∫ΩΓωx2dμω=∫ΩPωθx2dμω=∫Ωθxω2dμω=θx2≤θ2x2,∀x∈U.
Hence,Γωω∈Ω is a continuous g-Bessel sequence for U with respect to Vωω∈Ω. Now,(15)Kx=TΛθx=∫ΩΛω∗θxωdμω=∫ΩΛω∗Pωθxdμω,∀x∈U,showing that Γωω∈Ω is a continuous g-Bessel sequence.Proposition 3.
Suppose thatK∈ℬU and Λ=Λω∈ℬU,Vωω∈Ω is a continuous K-g-frame for U with respect to Vωω∈Ω; then, there exists a unique dual continuous K-Bessel sequence Γ=Γωω∈Ω of Λ such that TΓ≤Tθ for any dual continuous K-g-Bessel sequence Θ=θωω∈Ω of Λ.Proof.
SinceKK∗≤1/CΛTΛTΛ∗, by Lemma 2, it follows that there is a unique operator R∈ℬU,⊕ω∈ΩVw such that K=TΛR and(16)R2=infλ:K∗x2≤λTΛ∗x2,taking Γω=PωR for each ω∈Ω; then, as shown in Proposition 2, Γ is a continuous g-Bessel sequence for U with respect to Vωω∈Ω, and furthermore, it is a dual continuous K-g-Bessel sequence of Λ.
SincePωTΓ∗x=Γωx=PωRx for any ω∈Ω and x∈U, TΓ∗=R.
Now, letθωω∈Ω be any dual continuous K-g-Bessel sequence of Λωω∈Ω; then, K=TΛTθ∗, and K∗=TθTΛ∗ as a sequence. For each x∈U, we have(17)K∗x2=TθTΛ∗x2≤Tθ2TΛ∗x2.
It follows thatTθ2≥R2=TΓ∗2=TΓ2.
Equivalently,Tθ≥TΓ.Proposition 4.
Suppose thatK∈ℬU and Λ=Λωω∈Ω is a continuous K-g-frame for U with respect to Vωω∈Ω; then, the dual continuous K-g-Bessel sequence of Λ is precisely the family Γωω∈Ω=Pωθ∗ω∈Ω, where θ∈ℬ⊕ω∈ΩVw,U satisfies the condition K=TΛθ∗.Theorem 1.
LetK∈ℬU and Λ=Λωω∈Ω be a Parseval continuous K-g-frame for U with respect to Vωω∈Ω, where K has a closed range; then, ΛωK+∗ω∈Ω is a dual continuous K-g-Bessel sequence of Λ=Λωω∈Ω.Proof.
It is easy to see thatΛωK+∗ω∈Ω is a continuous K-g-frame for U with respect to Vωω∈Ω.
By Lemma1, y=K∗K∗+y=K∗K+∗y for every y∈ℛK∗; thus, by (5),(18)Ky=KK∗K+∗y=∫ΩΛω∗ΛωK+∗ydμω.
Ifh∈ℛK∗⊥=NK, then by Lemma 1 again, we obtain h∈NK∗+, implying that ∫ΩΛω∗ΛωK+∗hdμω=0=Kh. Altogether, we have(19)Kx=∫ΩΛω∗ΛωK+∗xdμω,∀x∈U,which ends the proof.Theorem 2.
LetK∈ℬU have a closed range and Λ=Λωω∈Ω be a Parseval continuous K-g-frame for U with respect to Vωω∈Ω. For each ω∈Ω, let Γω∈ℬU,Vw; then, Γ=Γωω∈Ω is a dual continuous K-g-Bessel sequence of Λ for upper bound BΓ if and only if there exists φ∈ℬU,⊕ω∈ΩVw such that TΛφ=0 and Γω=ΛωK+∗+Pωφ for each ω∈Ω. In this case, the continuous g-Bessel bound of Γ is K+K+φ2.Proof.
First, assume thatΓ is a dual continuous K-g-Bessel sequence of Λ, and we define φ:U⟶⊕ω∈ΩVw by φxω=Γωx−ΛωK+∗x for each x∈U and ω∈Ω; then, φ∈ℬU,⊕ω∈ΩVw. Indeed,(20)φx=φxωω∈Ω=Γωxω∈Ω−ΛωK+∗xω∈Ω≤Γωxω∈Ω+ΛωK+∗xω∈Ω=∫ΩΓωx21/2+∫ΩΛωK+∗x21/2≤BΓx+K∗K+∗x≤BΓ+KK+x,for each x∈U, and by Theorem 1, we have(21)TΛφx=∫ΩΛω∗φxωdμω=∫ΩΛω∗Γω−ΛωK+∗xdμω=∫ΩΛω∗Γωxdμω−∫ΩΛω∗ΛωK+∗xdμω=Kx−Kx=0.
Conversely, we show thatΓωω∈Ω is a continuous g-Bessel sequence for U with respect to Vωω∈Ω with bound K+K+φ2.
From Theorem1, we have(22)∫ΩΛω∗Γωxdμω=∫ΩΛω∗ΛωK+∗+Pωφxdμω=∫ΩΛω∗ΛωK+∗xdμω+TΛφx=∫ΩΛω∗ΛωK+∗xdμω=Kx,∀x∈U.Corollary 1.
LetK∈ℬU have a closed range and Λωω∈Ω be a Parseval continuous K-g-frame for U with respect to Vωω∈Ω, and let Γω∈ℬU,⊗ω∈ΩVw,∀ω∈Ω; then, Γωω∈Ω is a dual continuous K-g-Bessel sequence of Λ if and only if there exists a continuous g-Bessel sequence θωω∈Ω for U with respect to Vωω∈Ω such that(23)Γω=ΛωK+∗+θω,∀ω∈Ω,∫ΩΛω∗θωxdμω=0,∀x∈U.Proof.
Assume thatΓωω∈Ω is a dual continuous K-g-Bessel sequence of Λωω∈Ω; then, from Theorem 2, there exists φ∈ℬU,⊕ω∈ΩVw such that(24)Γω=ΛωK+∗+Pωφ,∀ω∈Ω,∫ΩΛω∗φxωx=0,∀x∈U.
Takingθω=Pωφ,∀ω∈Ω, then clearly, θωω∈Ω is a continuous g-Bessel sequence for U with respect to Vωω∈Ω with bound φ2, and(25)∫ΩΛω∗θωxdμω=∫ΩΛω∗φxωxdμω,∀x∈U.
Conversely, letΓωω∈Ω=ΛωK+∗+θωω∈Ω, where θωω∈Ω is a continuous g-Bessel sequence for U with respect to Vωω∈Ω such that(26)∫ΩΛω∗θωxdμω=0.
By Theorem1,(27)∫ΩΛω∗Γωxdμω=∫ΩΛω∗ΛωK+∗+θωxdμω=∫ΩΛω∗ΛωK+∗xdμω+∫ΩΛω∗θωxdμω=∫ΩΛω∗ΛωK+∗xdμω=Kx.Theorem 3.
LetK∈ℬU have a closed range and Λ=Λωω∈Ω be a Parseval continuous K-g-frame for U with respect to Vωω∈Ω; then, ΛωK+∗ is the canonical dual continuous K-g-Bessel sequence of Λ.Proof.
By Theorem1, we know that ΛωK+∗ is a dual continuous K-g-Bessel sequence of Λ; to complete this proof, we need to prove by Proposition 3 that TΛ′≤TΓ for any dual continuous K-g-Bessel sequence Γωω∈Ω of Λ, where TΛ′ is the synthesis operator of ΛωK+∗. By Theorem 2, there exists φ∈ℬU,⊕ω∈ΩVw such that TΛφ=0 and Γω=ΛωK+∗+Pωφ. A simple computation gives TΓ∗=TΛ′∗+φ, noting TΛ′=K+TΛ.
For eachx∈U, we have(28)TΓ∗x2=TΓ∗x,TΓ∗x=TΛ′∗x+φx,TΛ′∗x+φx=TΛ′∗x2+TΛ′∗x,φx+φx,TΛ′∗x+φx2=TΛ′∗x2+φx2≥TΛ′∗x2.
Therefore,TΛ′x=TΛ′∗x≤TΓ∗x=TΓx.The question of stability plays an important role in various fields of applied mathematics. The classical theorem of the stability of a base is due to Paley and Wiener. It is based on the fact that a bounded operatorT on a Banach space is invertible if we have I−T<1.Lemma 3 (see [9]; Paley–Wiener).
Letfii∈ℕ be a basis of a Banach space X and gii∈ℕ a sequence of vectors in X. If there exists a constant λ∈0,1 such that(29)∑i∈ℕcifi−gi≤λ∑i∈ℕcifi,for all finite sequences cii∈ℕ of scalars, then gii∈ℕ is also a basis for X.Proposition 5.
LetK∈ℬU have a closed range, Λωω∈Ω be a continuous K-g-frame for U with respect to Vωω∈Ω with bounds AΛandBΛ, and Γωω∈Ω be a dual continuous K-g-Bessel sequence of Λ with upper bound BΛ. For each ω∈Ω, let θω∈BU,Vω; if the following conditions are satisfied,(1)
λ=∫Ωθω−Λω2dμω<∞ and(2)
β=∫ΩK+Λω−θωΓdμω<1,then Θ=θωω∈Ω is a PLℛK continuous K-g-frame for U with respect to Vωω∈Ω with continuous K-g-frame bounds BΓ−1K+−21−β2 and λ+BΛ2, where L is defined by L:ℛK⟶U such that Lx=∫ωθω∗ΓωK+xdμω and PLℛK denotes the orthogonal projection on LℛK.Proof.
Define(30)ψ:⊕ω∈ΩVw⟶U,ψyωω∈Ω=∫ωθω∗yωdμω.
Then, (1) implies that ψ is well defined and bounded with ψ≤λ+BΛ.
Clearly,ψ is linear.
Thus,(31)∫ωθω∗yωdμω=ψ∗x2≤ψ2x2≤λ+BΛ2x2.
Therefore,θωω∈Ω is a continuous K-g-Bessel sequence for U with respect to Vωω∈Ω.
For anyx∈ℛK, we have(32)x−Lx=∫ωΛω∗ΓωK+xdμω−∫ωθω∗ΓωK+xdμω=∫ωΛω∗−θω∗ΓωK+xdμω≤∫ωK+Λω−θωΓωxdμω=βx.
Hence,(33)1−βx≤Lx,∀x∈ℛK.
From this, we conclude that the operatorL:ℛK⟶LℛK is invertible with L−1≤1/1−β. It is trivial to show that LℛK is closed.
For anyx∈U, we have(34)PLℛKKx=LL−1PLℛKKx=∫Ωθω∗ΓωK+L−1PLℛKKxdμω.
Therefore,(35)K∗PLℛKx2=supy=1K∗PLℛKx,y2=supy=1∫ωθω∗ΓωK+L−1PLℛKKx,y2=supy=1∫ΩΓωK+L−1PLℛKKx,θωy2≤∫Ωθωx2dμωsupy=1∫ΩΓωK+L−1PLℛKKy2dμωn≤BΓK+2L−1K2∫Ωθωx2dμωn≤BΓK+2K211−β2∫Ωθωx2dμω.
It follows that(36)BΓ−1K+−2K−2K∗PLℛKx2≤∫Ωθωx2dμω.
This completes the proof.


## 3. ContinuousK-g-Frame Sequences and Continuous g-Frame Sequences

Theorem 4.
LetΛ=Λωω∈Ω be a g-frame for U with respect to Vωω∈Ω; then, Λ is a continuous K-g-frame for U with respect to Vωω∈Ω if and only if ℛK⊂span¯Λω∗Vωω∈Ω.Proof.
Suppose thatΛ is a continuous K-g-frame for U with respect to Vωω∈Ω; then, ∀x∈U, we have(37)CΛKK∗x,x=CΛK∗x2≤∫ΩΛωxdμω=TΛ∗x2=TΛTΛ∗x,x.
From Lemma2, we have ℛK⊂ℛTΛ since ℛTΛ⊆span¯Λω∗Vωω∈Ω. Conversely, let ℛK⊂span¯Λω∗Vωω∈Ω.
SinceU=span¯Λω∗Vωω∈Ω⊕span¯Λω∗Vωω∈Ω⊥, we have(38)∀x∈U,x=y+z;y∈span¯Λω∗Vωω∈Ω,z∈span¯Λω∗Vωω∈Ω⊥.
For anyω∈Ω,Λω∗Λωz∈span¯Λω∗Vωω∈Ω.
For allx∈U, Λωx2=Λωy2, so(39)∫ΩΛωx2dμω=∫ΩΛωy2dμω≤BΛy2+z2=BΛx2.
On the contrary, sinceℛK⊂span¯Λω∗Vωω∈Ω, z∈ℛK⊥; then, K∗z,h=z,Kh=0,h∈U; hence, K∗z=0.
Then,(40)∫ΩΛωx2dμω=∫ΩΛωy2dμω≥CΛK∗y2=CΛK∗x2.Proposition 6.
LetK∈ℬU have a closed range and Λ=Λωω∈Ω be a continuous K-g-frame for U with respect to Vωω∈Ω; then, Λ=Λωω∈Ω is a continuous g-frame for ℛK with respect to Vωω∈Ω.Proof.
We just prove the lower continuous g-frame inequality. From Proposition2, there exists a continuous g-Bessel sequence Γ=Γωω∈Ω for U with respect to Vωω∈Ω such that Kx=∫ΩΛω∗Γωx,∀x∈U.
For everyy∈ℛK, by Lemma 1,(41)y=KK+y=∫ΩΛω∗ΓωK+ydμω=∫ΩΛω∗ΓωK+PℛKydμω.
Takeθω=ΓωK+PℛK for every ω∈Ω.
Then, we have(42)∫Ωθωx2dμω=∫ΩΓωK+PℛK2dμω≤BΓK+2x2.
So,θωω∈Ω is a continuous g-Bessel sequence for U with respect to Vωω∈Ω.
For anyy,z∈ℛK, we have(43)y,z=∫ΩΛω∗θωydμω,z=∫ΩΛω∗θωy,zdμω=∫ΩΓωK+PℛKy,Λwzdμω=∫Ωy,ΓωK+PℛK∗,Λωzdμω=y,∫ΩΓωK+PℛK∗Λωzdμω=y,∫ΩθωΛωzdμω,and then z=∫ΩθωΛωzdμω.
Hence,(44)z2=supy=1z,y2=supy=1∫ΩθωΛωzdμω,y2≤supy=1∫Ωθωy2dμω∫ΩΛωz2dμω≤BΓK+2∫ΩΛωz2dμω.
Then,Λωω∈Ω is a continuous g-frame for ℛK with respect to Vωω∈Ω.Proposition 7.
LetK∈ℬU have a closed range and Λ=Λωω∈Ω be a continuous K-g-frame for ℛK with respect to Vωω∈Ω. If span¯Λω∗Vωω∈Ω⊂ℛK, then Λ=Λωω∈Ω is a continuous K-g-frame for U with respect to Vωω∈Ω.Proof.
For eachx∈U, we have x=y+z such that y∈ℛK and z∈ℛK⊥. So, Λωx2=Λωy2.
Sincespan¯Λω∗Vωω∈Ω⊂ℛK and Λω∗Λωe∈span¯Λω∗Vωω∈Ω,∀e∈U,(45)∫ΩΛx2dμω=∫ΩΛy2dμω≤BΛx2≤BΛy2+z2=BΛx2.
LetΦ=ϕωω∈Ω be a dual continuous g-frame of Λ=Λωω∈Ω.
So,(46)∫ΩΛω∗ϕωydμω=∫Ωϕω∗Λωydμω.
Hence,(47)K∗x2=K∗y+z=K∗y2=K∗y,K∗y=KK∗y,y=∫Ωϕω∗Λωydμω,KK∗y=∫ΩΛωy,ϕωKK∗ydμω≤∫ΩΛωy2dμω1/2∫ΩϕωKK∗y2dμω1/2≤BϕKK∗y∫ΩΛωy21/2≤BϕKK∗x∫ΩΛωy2dμω1/2.
Then,Bϕ−1K−2K∗x2≤∫ΩΛωy2.
On the contrary, we have∫ΩΛωy2=∫ΩΛωx2.
Then,Bϕ−1K−2K∗x2≤∫ΩΛωx2.
So,Λ is a continuous K-g-frame for U.Theorem 5.
For everyω∈Ω, let Λω∈ℬU,Vw; then, the following statements are equivalent:(1)
Λ=Λωω∈Ω is a continuous K-g-frame sequence for U with respect to Vωω∈Ω.(2)
Λ=Λωω∈Ω is a continuous g-Bessel sequence for span¯Λω∗Vωω∈Ω with respect to Vωω∈Ω, and there exists a continuous g-Bessel sequence Γωω∈Ω for U with respect to Vωω∈Ω such that(48)K∗span¯Λω∗Vωω∈Ω∗x=∫ΩΛω∗Γωx,∀x∈U.(3)
Λ=Λωω∈Ω is a continuous g-Bessel sequence for span¯Λω∗Vωω∈Ω with respect to Vωω∈Ω, and there exists a continuous g-Bessel sequence Γωω∈Ω for U with respect to Vωω∈Ω such that(49)K∗x=∫ΩΓω∗Λωxdμω,∀x∈span¯Λω∗Vωω∈Ω.Proof.
1⟹2:We have(50)K∗span¯Λω∗Vωω∈Ω:span¯Λω∗Vωω∈Ω⟶Uthen their adjoint(51)K∗span¯Λω∗Vωω∈Ω∗:U⟶span¯Λω∗Vωω∈Ω.By the definition on the continuousK-g-frame sequence, there exists CΛ>0 such that(52)CΛK∗x2≤TΛ∗x2,∀x∈span¯Λω∗Vωω∈Ω.This implies(53)CΛK∗span¯Λω∗Vωω∈Ω∗K∗span¯Λω∗Vωω∈Ω≤TΛTΛ∗.From Lemma2, there exists R∈ℬU,⊕ω∈ΩVw such that(54)K∗span¯Λω∗Vωω∈Ω∗=TΛR.For everyω∈Ω, denote Γω=PωR; then, we have(55)∫ΩΓωx2dμω=∫ΩPωRx2dμω=∫ΩRxω2dμω=Rx2≤R2x2.Hence,Γωω∈Ω is a continuous g-Bessel sequence for U with respect to Vωω∈Ω.On the contrary, we have for allx∈U,(56)K∗span¯Λω∗Vωω∈Ω∗x=TΛRx=∫ΩΛω∗Rxωdμω=∫ΩΛω∗PωRxdμω=∫ΩΛω∗Γωxdμω.2⟹3:For everyx∈U and y∈span¯Λω∗Vωω∈Ω, by (48), we have(57)K∗span¯Λω∗Vωω∈Ω∗x,y=∫ΩΛω∗Γωx,ydμω=x,∫ΩΓω∗Λωydμω.So,(58)x,K∗y=x,K∗span¯Λω∗Vωω∈Ωy=x,∫ΩΓω∗Λωydμω.Hence,K∗y=∫ΩΓω∗Λωydμω.3⟹1:
Suppose that (49) holds; to prove (1), we just prove the lower bound inequality on continuous K-g-frames.
For everyy∈span¯Λω∗Vωω∈Ω, we have(59)K∗y2=supz=1K∗y,z2=supz=1∫ΩΓω∗Λωydμω,z2=supz=1∫ΩΛωydμω,Γωz2≤∫ΩΛωy2dμωsupz=1∫ΩΓωz2dμω≤BΓ∫ΩΛωy2dμω.
So,BΛ−1K∗y2≤∫ΩΛωy2dμω,∀y∈span¯Λω∗Vωω∈Ω.Theorem 6.
A sequenceΛωω∈Ω∈ℬU,⊕ω∈ΩVw is a continuous K-g-frame sequence for U with respect to Vωω∈Ω if and only if there exists T∈ℬspan¯Λω∗Vωω∈Ω,⊕ω∈ΩVw such that(60)Λωspan¯Λω∗Vωω∈Ω=PωT,∀ω∈Ω,ℛK∗span¯Λω∗Vωω∈Ω⊂ℛT∗.Proof.
LetΛωω∈Ω be a continuous K-g-frame sequence for U with respect to Vωω∈Ω, and let T=TΛ∗.
Hence,Λω/span¯Λω∗Vωω∈Ωx=PωTΛ∗x=PωTx for every x∈span¯Λω∗Vωω∈Ω and any ω∈Ω. So, Λω/span¯Λω∗Vωω∈Ω=PωT.
From (53), we have(61)CΛK∗span¯Λω∗Vωω∈Ω∗K∗span¯Λω∗Vωω∈Ω≤T∗T.
By Lemma2, it follows that ℛK∗/span¯Λω∗Vωω∈Ω∗⊂ℛT∗.
Conversely,Λω/span¯Λω∗Vωω∈Ω=PωT for any ω∈Ω.
Letx∈span¯Λω∗Vωω∈Ω; we have(62)∫ΩΛωx2dμω=∫ΩΛωspan¯Λω∗Vωω∈Ωx2dμω=∫ΩPωT2dμω=Tx2≤T2x2.
So,Λωω∈Ω is a continuous K-g-frame sequence for span¯Λω∗Vωω∈Ω with respect to Vωω∈Ω.
SinceℛK∗/span¯Λω∗Vωω∈Ω∗⊂ℛT∗, by Lemma 2, there exists λ>0 such that(63)K∗span¯Λω∗Vωω∈Ω∗K∗span¯Λω∗Vωω∈Ω≤λT∗T.
Hence, for everyx∈span¯Λω∗Vωω∈Ω, we have(64)λ−1K∗x2=λ−1K∗span¯Λω∗Vωω∈Ωx2≤Tx2=∫ΩΛx2dμω,and then Λωω∈Ω is a continuous K-g-frame sequence for span¯Λω∗Vωω∈Ω with respect to Vωω∈Ω with bounds λ−1 and T2.In general, a continuousK-g-frame Λωω∈Ω for U with respect to Vωω∈Ω is not a continuous g-frame for span¯Λω∗Vωω∈Ω with respect to Vωω∈Ω.Proposition 8.
LetΛ=Λωω∈Ω be a continuous K-g-frame sequence for U with respect to Vωω∈Ω. Suppose that K∗/span¯Λω∗Vωω∈Ω≠0 and that it has a closed range; then, Λωω∈Ω is a continuous g-frame for ℛK∗/span¯Λω∗Vωω∈Ω∗ with respect to Vωω∈Ω.Proof.
SinceK∗/span¯Λω∗Vωω∈Ω has a closed range, its pseudo-inverse exists; from Lemma 1, every x∈ℛK∗/span¯Λω∗Vωω∈Ω∗ can be written as(65)x=K∗span¯Λω∗Vωω∈Ω+K∗span¯Λω∗Vωω∈Ωx.
Hence,(66)x2≤K∗span¯Λω∗Vωω∈Ω+2K∗span¯Λω∗Vωω∈Ωx2.
Noting thatℛK∗/span¯Λω∗Vωω∈Ω∗⊂span¯Λω∗Vωω∈Ω, we have(67)K∗span¯Λω∗Vωω∈Ωx=K∗x.
So,(68)x2≤K∗span¯Λω∗Vωω∈Ω+2K∗x2≤1AΛK∗span¯Λω∗Vωω∈Ω+2∫ΩΛωx2dμω.
SinceK∗/span¯Λω∗Vωω∈Ω≠0, its pseudo-inverse(69)K∗span¯Λω∗Vωω∈Ω+≠0,and then(70)AΛK∗span¯Λω∗Vωω∈Ω+−2x2≤∫ΩΛωx2dμω.
On the contrary, we have(71)∫ΩΛωx2dμω≤BΛx2,∀x∈ℛK∗span¯Λω∗Vωω∈Ω∗.
So,Λωω∈Ω is a continuous g-frame for ℛK∗/span¯Λω∗Vωω∈Ω∗ with respect to Vωω∈Ω with bounds CΛK∗/span¯Λω∗Vωω∈Ω+−2 and BΛ.In the following, we give a necessary and sufficient condition under which a tight continuousK-g-frame sequence Λωω∈Ω for U with respect to Vωω∈Ω is a tight continuous g-frame for span¯Λω∗Vωω∈Ω with respect to Vωω∈Ω.Proof.
Theorem 7LetΛωω∈Ω be a C-tight continuous g-frame for U with respect to Vωω∈Ω; then, Λωω∈Ω is a D-tight continuous g-frame for span¯Λω∗Vωω∈Ω with respect to Vωω∈Ω if and only if K∗/span¯Λω∗Vωω∈Ω admits a left inverse C/DK∗/span¯Λω∗Vωω∈Ω∗.Proof.
First, we suppose thatΛωω∈Ω is a D-tight continuous g-frame for span¯Λω∗Vωω∈Ω with respect to Vωω∈Ω.
Hence,(72)Dx2=∫ΩΛωx2dμω,K4∀x∈span¯Λω∗Vωω∈Ω,CK∗x2=∫ΩΛωx2dμω,∀x∈span¯Λω∗Vωω∈Ω.
So,Dx2=CK∗x2 for every x∈span¯Λω∗Vωω∈Ω. Then,(73)Dx,x=CK∗x,K∗x=CK∗span¯Λω∗Vωω∈Ω∗K∗span¯Λω∗Vωω∈Ωx,x.
This implies that(74)CDKspan¯Λω∗Vωω∈Ω∗∗K∗span¯Λω∗Vωω∈Ω=Idspan¯Λω∗Vωω∈Ω.
Hence,C/DK∗/span¯Λω∗Vωω∈Ω∗ is a left inverse of K∗/span¯Λω∗Vωω∈Ω; conversely, we have(75)x=CDK∗span¯Λω∗Vωω∈Ω∗.
So,(76)∫ΩΛωx2dμω=CK∗x2=DCDK∗x2=DCDK∗span¯Λω∗Vωω∈Ωx,K∗span¯Λω∗Vωω∈Ωx=Dx2.
It follows thatΛ=Λωω∈Ω is a D-tight continuous g-frame for span¯Λω∗Vωω∈Ω with respect to Vωω∈Ω.Theorem 7.
LetK1,K2∈ℬU and Λωω∈Ω be a C-tight continuous K1-g-frame sequence for U with respect to Vωω∈Ω; then, Λωω∈Ω is a C-tight continuous K2-g-frame sequence for U with respect to Vωω∈Ω if and only if(77)ℛK2∗span¯Λω∗Vωω∈Ω∗⊂ℛK1∗span¯Λω∗Vωω∈Ω∗.Proof.
First, we suppose thatΛ=Λωω∈Ω is a C-tight continuous K2-g-frame sequence for U with respect to Vωω∈Ω; then, there exists D>0 such that(78)DK2∗x2≤∫ΩΛωx2dμω,∀x∈span¯Λω∗Vωω∈Ω.
We haveΛ=Λωω∈Ω is a C-tight continuous K1-g-frame sequence for U with respect to Vωω∈Ω; then, K2∗x2≤C/DK1∗x2 for every x∈span¯Λω∗Vωω∈Ω.
Hence,(79)K2∗span¯Λω∗Vωω∈Ωx,K2∗span¯Λω∗Vωω∈Ωx=K2∗x,K2∗x≤CDK1∗x,K1∗x=CDK1∗span¯Λω∗Vωω∈Ωx,K1∗span¯Λω∗Vωω∈Ωx.
Therefore,(80)K2∗span¯Λω∗Vωω∈Ω∗K2∗span¯Λω∗Vωω∈Ω≤CDK1∗span¯Λω∗Vωω∈ΩK1∗span¯Λω∗Vωω∈Ω.
By Lemma2, we have(81)ℛK2∗span¯Λω∗Vωω∈Ω∗⊂ℛK1∗span¯Λω∗Vωω∈Ω∗.
Conversely, supposeℛK2∗/span¯Λω∗Vωω∈Ω∗⊂ℛK1∗/span¯Λω∗Vωω∈Ω∗.
By Lemma2, there exists λ>0 such that(82)K2∗span¯Λω∗Vωω∈Ω∗K2∗span¯Λω∗Vωω∈Ω≤λK1∗span¯Λω∗Vωω∈ΩK1∗span¯Λω∗Vωω∈Ω,so for every x∈span¯Λω∗Vωω∈Ω, we have(83)K2∗x2=K2∗span¯Λω∗Vωω∈Ω∗K2∗span¯Λω∗Vωω∈Ωx,x≤λK1∗span¯Λω∗Vωω∈ΩK1∗span¯Λω∗Vωω∈Ω=λK1∗x2=λc∫ΩΛωx2dμω.
Then, for allx∈span¯Λω∗Vωω∈Ω,(84)cλK2∗x2≤∫ΩΛωx2dμω≤CK1∗2x2.
This completes the proof.


## 4. Conclusion

In this article, we have proved some properties of dual continuousK-g-frames in Hilbert spaces. These results are extensions of the related results announced in [8]. The presented theorems extend, generalize, and improve many existing results in the literature.



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