# Characterization, Dilation, and Perturbation of Basic Continuous Frames

Authors: Xin Zhao; Pengtong Li
Journal: Journal of Mathematics
 (2022)

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## Abstract

A vector-valued function is called a basic continuous frame if it is a continuous frame for its spanning space. It is shown in this article that basic continuous frames and their oblique duals can be characterized by operators with closed ranges. Furthermore, we show that any oblique dual pair of basic continuous frames for a Hilbert space can be dilated to a Type II dual pair for a larger Hilbert space. Finally, a perturbation result for basic continuous frames is given. Since the spanning spaces of two basic continuous frames for a Hilbert space are often different, the research process is more complex than the setting of general continuous frames.


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## Body

## 1. Introduction

The concept of discrete frames was first formally introduced by Duffin and Schaeffer [1] in 1952 and popularized greatly after the significant paper [2] by Daubechies et al. in 1986. A discrete frame is an overcomplete family of countable elements in a Hilbert space which allows every element in the space to be represented as a linear combination of the frame elements. It has been widely used in many fields such as image and signal processing, approximation theory and wireless communications. Due to different applications and theoretical goals, various generalizations of discrete frames have been presented. For example, pseudoframes [3], g-frames [4], fusion frames [5] and operator-valued frames [6]. One important generalization of discrete frames is the so-called continuous frames, which is introduced by Kaiser [7] and independently by Ali, Antonie and Gazeau [8]. We refer to [9–12] for more studies on continuous frames.Dilation and perturbation are two significant properties for discrete and continuous frames. Gabardo and Han [13] generalized the dilation theorem for dual discrete frames (cf. [14]) to dual continuous frames. Using the method considered by Casazza and Christensen (cf. [15]), they deduced a perturbation theorem for continuous frames. Kaushik et al. [16] gave some equivalent conditions of perturbation for continuous frames and obtained a sufficient condition for the stability of a continuous frame. Since the basic continuous frame is more general in theory and more freedom when the dual frame is selected, we study basic continuous frames around their dilation and perturbation properties.Throughout this paper,ℋ and K denote Hilbert spaces over the complex field and J a countable index set. For an operator T∈Bℋ,K, we denote its range and kernel by ranT and kerT, respectively. Let U, V be closed subspaces of ℋ. We denote by U∔V the sum of two subspaces with trivial intersection, by U⊕V the orthogonal direct sum of closed subspaces and by U⊖V the space U∩V⊥. We call an operator P∈Bℋ a (oblique) projection if it satisfies P2=P. In this case, the decomposition ℋ=ranP∔kerP holds. Conversely, if ℋ=U∔V, then we can find a unique projection PU,V satisfying ranPU,V=U and kerPU,V=V. Let PU:=PU,U⊥ be the orthogonal projection onto U and PU|V be its restriction to V.LetΩ,μ be a measure space where μ is positive. Recall that a vector-valued function F:Ω⟶ℋ is said to be a continuous frame for ℋ with respect to Ω,μ or a Ω,μ-frame if(i)
F is weakly measurable, i.e., for all x∈ℋ, ω⟶x,Fω is a measurable function on Ω.(ii)
There exist constantsA, B>0 such that(1)Ax2≤∫Ωx,Fω2dμω≤Bx2,∀x∈ℋ.The numbersA, B are called frame bounds. We call F a Ω,μ-Bessel mapping if only the second inequality of (1) holds, a tight Ω,μ-frame if A=B, and a Parseval Ω,μ-frame if A=B=1. The mapping F is called a basic Ω,μ-frame or a basic continuous frame for ℋ if it is a continuous frame for the spanning spacespan¯Fωω∈Ω. When Ω=ℕ and μ is the counting measure, the family Fωω∈Ω is a discrete frame.To aΩ,μ-Bessel mapping F, we associate the analysis operatorθF given by(2)θF:ℋ⟶L2Ω,μ,θFxω≔x,Fω,Note that the frame condition (1) implies that Ax≤θFx≤Bx. The adjoint TF of θF is called synthesis operator and is given by(3)TF:L2Ω,μ⟶ℋ,TFf,y=∫ΩfωFω,ydμω,∀y∈ℋ.In this case, we usually say that the following equation holds in the weak sense:(4)TFf=∫ΩfωFωdμω.Theframe operatorSF is defined to be TFθF. When F is a continuous frame for ℋ, SF is a bounded, invertible and positive operator. A Ω,μ-Bessel mapping G is said to a dual of F if TFθG=Iℋ, i.e.,(5)x=∫Ωx,GωFωdμω,∀x∈ℋ.EveryΩ,μ-frame always has a dual SF−1F, called the canonical dual, satisfying the following reconstruction formula:(6)x=∫Ωx,SF−1FωFωdμω,∀x∈ℋ.IfF has the unique dual, then we say F is a Riesz-typeΩ,μ-frame. We see from ([13], Proposition 1) that F is a Riesz-type Ω,μ-frame if and only if its analysis operator is surjective. Two Ω,μ-frames F and G for ℋ and K, respectively, are called similar if LF=G holds for an invertible operator L:ℋ⟶K. Let ℳ be a closed subspace of L2Ω,μ. We call M a frame range if there exist a Hilbert space ℋ and a Ω,μ-frame F for ℋ whose analysis operator has range space ℳ.The paper is organized as follows. In Section2, we present some preliminary results about basic Ω,μ-frames. In Section 3, we characterize basic Ω,μ-frames and their oblique duals. Section 4 is devoted to the dilation of oblique dual pairs of basic Ω,μ-frames. A perturbation theorem of basic Ω,μ-frames is considered in Section 5.


## 2. Preliminaries

This section is devoted to some preliminary results about basic continuous frames and their duals. In the rest of the paper, we agree to use the following notation:(7)ℋF≔span¯Fωω∈Ω.For a given vector-valued functionF:Ω⟶ℋ. For a Ω,μ-Bessel mapping F, we always use θF and TF to denote the analysis operator and synthesis operator, respectively.Recall that every basic continuous frame is aΩ,μ-Bessel mapping. Conversely, a Ω,μ-Bessel mapping F constitutes a basic continuous frame if and only if ranTF is closed. In this case, ℋF=ranTF.For a basicΩ,μ-frame F, the frame operator SF is invertible when restricted to span¯Fωω∈Ω. Its canonical dual is F˜=SF†F, where SF† denotes the pseudoinverse of SF (see [17]). We recall that(8)span¯Fωω∈Ω=span¯F˜ωω∈Ω,ranθF=ranθF˜.Now, we give definitions on duals of basicΩ,μ-frames with the following. For the discrete case, we see [18–20]Definition 1.
Suppose thatF is a basic Ω,μ-frame and that G is a Ω,μ-Bessel mapping for ℋ.(i)
G is a dual of F if TFθGℋF=IℋF, i.e., x=∫Ωx,GωFωdμω, ∀x∈ℋF(ii)
G is a Type I dual of F if G is a dual of F and ranTG⊂ranTF(iii)
G is a Type II dual of F if G is a dual of F and ranθG⊂ranθF(iv)
G is an oblique dual of F if G is a basic Ω,μ-frame that is a dual of F and F is a dual of G
Heil et al. showed in [20] that Type I or II duals of discrete frame sequences are oblique duals. They also characterized the existence of oblique duals with respect to the direct sum decomposition of ℋ. It is easy to show that these facts still hold for basic Ω,μ-frames. If G is a Type I dual of F, then G is an oblique dual of F and ranTG=ranTF; if G is a Type II dual of F, then G is an oblique dual of F and ranθG=ranθF. Similar to ([20], Theorem 1.4), we also have:Proposition 1.
Suppose thatU and V are closed subspaces of ℋ and that F is a Ω,μ-frame for U. Then the following are equivalent:(i)
ℋ=U∔V(ii)
There is aΩ,μ-frame G for V that is a Type II dual of F(iii)
There is aΩ,μ-frame G for V that is an oblique dual of F
From the above proposition, the fact thatG is an oblique dual of F shows ℋ=ℋF∔ℋG⊥. For a pair of oblique duals, L2Ω,μ also have a decomposition.Lemma 1.
Suppose thatF is a basic Ω,μ-frame for ℋ and that G is an oblique dual of F. Assume ℳ⊂L2Ω,μ is closed containing both ranθF and ranθG. Then,(i)
L2Ω,μ=ranθF∔ranθG⊥=ranθG∔ranθF⊥(ii)
ℳ=ranθF∔ℳ⊖ranθG=ranθG∔ℳ⊖ranθFProof.
(1) By assumption, we have(9)TGθF|ℋG=I|ℋG,ℋ=ℋG∔ℋF⊥=ℋF∔ℋG⊥.
Hence,(10)θFTG2=θFTGθFTG=θFTG,which implies that θFTG is a projection. Using (9), we obtain(11)ranθF=θFℋG∔ℋF⊥=θFℋG=ranθFTG,ranθG=θGℋF∔ℋG⊥=θGℋF=ranθGTF.
Therefore,kerθFTG=ranθGTF⊥=ranθG⊥ and thus θFTG=PranθF,ranθG⊥. This implies that(12)L2Ω,μ=ranθFTG∔kerθFTG=ranθF∔ranθG⊥.
(ii). Note that (i) shows thatranθF∩ℳ⊖ranθG=0. Now, the conclusion follows from ([13], Lemma 3.9)Li et al. [21] gave the decomposition of frame ranges in terms of the decomposition of analysis operators. They showed that not all closed subspaces of L2Ω,μ constitute frame ranges:Lemma 2 (see [21]).
LetΩ,μ be a σ-finite, positive measure space.(i)
L2Ω,μ itself is a frame range if and only if Ω,μ is purely atomic(ii)
Every closed subspace of a frame range is a frame rangeWe finish this section with the following equivalent condition about the frame range:Lemma 3 (see [13], Corollary 2.9).
For a closed subspaceℳ of L2Ω,μ, the following are equivalent:(i)
ℳ is a frame range(ii)
There exists an orthonormal basisψjj∈J for ℳ such that ∑j∈Jψjω2<∞ for a.e. ω∈Ω


## 3. Characterization of Basic Continuous Frames and Their Oblique Duals

This section focuses on the characterization of basic continuous frames and their oblique duals. We begin with a characterization for basic parsevalΩ,μ-frames.Lemma 4.
Suppose thatΩ,μ is a measure space with positive measure μ. Then the following are equivalent:(i)
Fis a basic parseval cotinuous frame forℋ(ii)
Fω=∑j∈Jψjω¯θ∗ψj for some partial isometry θ:ℋ⟶L2Ω,μ whose range is a frame range and some orthonormal basis ψjj∈J of ranθ(iii)
Fω=∑j∈Jψjω¯ej for some orthonormal set ejj∈J of ℋ and some orthonormal basis ψjj∈J of a frame range ℳProof.
(i)⇒ (ii). Let θ be the analysis operator for F and ℳ=ranθ. Since F is a basic parseval continuous frame, θ is a partial isometry. By Lemma 3, we can denote an orthonormal basis ψjj∈J for j∈J satisfying ∑ℳψjω2<∞ for a.e. ω∈Ω. For any x∈ℋ, it follows that(13)x,∑j∈Jψjω¯θ∗ψj=∑j∈Jθx,ψjψjω=θxω=x,Fω,which implies Fω=∑j∈Jψjω¯θ∗ψj.
(ii)⇒ (iii). Write ℳ=ranθ and ej=θ∗ψj for every j∈J. Since θ is a partial isometry, θ∗ is an isometry restricted to kerθ∗⊥=ranθ=ℳ. Hence ejj∈J is an orthonormal set.
(iii)⇒ (i). For any x∈span¯Fωω∈Ω⊂span¯ejj∈J, it follows that(14)∫Ωx,Fω2dμω=x,Fω2=∑j∈Jx,ejψjω2=∑j∈Jx,ej2=x2,implying that F is a basic parseval continuous frame.Similarly, we can deduce the following characterization for basicΩ,μ-frames.Proposition 2.
Suppose thatΩ,μ is a measure space with positive measure μ. Then the following are equivalent:(i)
F is a basic Ω,μ-frame for ℋ(ii)
Fω=∑j∈Jψj∈Jω¯θ∗ψj for some operator θ:ℋ⟶L2Ω,μ whose range is a frame range and some orthonormal basis of ranθ(iii)
Fω=∑j∈Jψjω¯ej for some Riesz sequence ejj∈J of ℋ and some orthonormal basis ψjj∈J of a frame range ℳProof.
(i)⇒ (ii) follows from the proof of Lemma 4. For (ii) ⇒ (iii), we see that θ∗ is bijective restricted to ℳ⟶ranθ∗. Hence, θ∗ψjj∈J is a Riesz sequence for ℋ.
Now, suppose (iii) holds. For anyx∈span¯Fωω∈Ω⊂span¯ejj∈J, it follows that(15)∫Ωx,Fω2dμω=x,Fω2=∑j∈Jx,ejψjω2=∑j∈Jx,ej2.
The implication (iii)⇒ (i) now follows from the fact that ejj∈J is a Riesz sequence. □
Suppose nowℳ is a frame range and that ψjj∈J is an orthonormal basis for ℳ. We compute(16)∫Ωφ,∑j∈Jψjω¯ψj2dμω=φ,∑j∈Jψjω¯ψj2=∑j∈Jφ,ψjψjω2=φ2,∀φ∈ℳ,implying that ω⟶∑j∈Jψjω¯ψj forms a parseval Ω,μ-frame for ℳ. If F is a basic Ω,μ-frame for ℋ with associated analysis operator θ, then it follows from Proposition 2 that(17)Fω=∑j∈Jψjω¯θ∗ψj=θ∗∑j∈Jψjω¯ψj,where θ∗ is invertible restricted to ℳ⟶ranθ∗. This means that every basic Ω,μ-frame is similar to a basic parseval Ω,μ-frame for L2Ω,μ.
Putting Lemma2 and Proposition 2 together, we can characterize basic Riesz-type Ω,μ-frames:Corollary 1.
Suppose thatΩ,μ is a purely atomic, positive measure space where μ is σ-finite, and that ψjj∈J is an orthonormal basis for L2Ω,μ. Then, the following are equivalent:(i)
F is a basic Riesz-type Ω,μ-frame for ℋ(ii)
Fω=∑j∈Jψjω¯θ∗ψj for some surjective operator θ:ℋ⟶L2Ω,μ(iii)
Fω=∑j∈Jψjω¯ej for some Riesz sequence ejj∈J of ℋ
It is known that there is a bijective correspondence between the set of discrete frame sequences and all the operators with closed range. The following proposition derives a corresponding result for basic continuous frames. This is the essential difference between discrete frame sequences and basic continuous frames.Proposition 3.
Every basicΩ,μ-frame corresponds to an operator whose range is a frame range.Proof.
Supposeθ:ℋ⟶L2Ω,μ is an operator whose range is a frame range. Then, by Proposition 2, Fω=∑j∈Jψjω¯θ∗ψj is a basic continuous frame for ℋ, where ψj∈J is an orthonormal basis for ranθ. Moreover, we compute(18)x,Fω=∑j∈Jθx,ψjψjω=θxω,∀x∈ℋ,which implies that θ is the analysis operator for F.A new basicΩ,μ-frame can be obtained by applying a suitable operator to a basic Ω,μ-frame. Note that the spanning space may be changed after the action of an operator. Using Proposition 3, we can derive it in an easy way.Corollary 2.
Suppose thatF is a basic Ω,μ-frame where μ is σ-finite, and that W is an operator on ℋ. Then, WF is a basic continuous frame if and only if W has a closed range restricted to span¯Fωω∈Ω.Proof.
By assumption,WF is a Ω,μ-Bessel mapping with associated analysis operator θFW∗.
Now, supposeWF is a basic continuous frame. Then, the analysis operator θFW∗ has a closed range and so is WθF∗. This means that W has a closed range restricted to ranθF∗=span¯Fωω∈Ω.
Conversely, supposeW has a closed range. Since ranθFW∗⊂ranθF and ranθF is a frame range, we see from Lemma 2 (ii) that ranθFW∗ is also a frame range. Then, by Proposition 3, WF is a basic continuous frame.We finish this section with the following characterization for an oblique dual pair of basicΩ,μ-frames.Proposition 4.
ForΩ,μ-Bessel mappings F and G of ℋ, the following are equivalent:(i)
G is an oblique dual of F(ii)
TFθG=PℋF,ℋG⊥(iii)
TGθF=PℋG,ℋF⊥(iv)
TFθGℋF=IℋF, ℋ=ℋF∔ℋG⊥(v)
TGθFℋG=IℋG, ℋ=ℋG∔ℋF⊥(vi)
θFTG=PranθF,ranθG⊥, where F, G are basic Ω,μ-frames(vii)
θGTF=PranθG,ranθF⊥ where F, G are basic Ω,μ-framesProof.
(i)⇒ (ii). By the definition of oblique duals, we get that(19)TFθGℋF=IℋF,TGθFℋG=IℋG.
Hence, we have(20)TFθG2=TFθG,ranTFθG=ℋFand ranTGθF=ℋG,which implies that TFθG is an oblique projection with ranTFθG=ℋF and kerTFθG=ranTGθF⊥=ℋG⊥.
(ii)⇒ (i). Since TFθG=PℋF,ℋG⊥, we get(21)x2=TFθGx,x=∫Ωx,GωFω,xdμω,∀x∈ℋF.
Using Cauchy Schwarz’ inequality and thatG is a Ω,μ-Bessel mapping, we can deduce the lower frame condition for F. Therefore F is a Ω,μ-frame for ℋF and G is a dual of F. We can prove G is a Ω,μ-frame and F is a dual of G in the same way.
(ii)⇒ (iv). The existence of PHF,ℋG⊥ implies that ℋ=ℋF∔ℋG⊥.
(iv)⇒ (ii). For any x∈ℋ, we can write x=y+z, where y∈ℋF, z∈ℋG⊥. Hence(22)TFθGx=TFθGy=y=PℋF,ℋG⊥x,
which impliesTFθG=PℋF,ℋG⊥.
(i)⇒ (vi) follows from the proof of Lemma 1.
(vi)⇒ (ii). Since θFTG=PranθF,ranθG⊥, we have(23)θGTF=PranθG,ranθF⊥,L2Ω,μ=ranθG∔ranθF⊥.
From the identity(24)TFθGTF=TFPranθG,ranθF⊥=PranθF,ranθG⊥θF∗=TF,it follows that(25)TFθG2=TFθGTFθG=TFθG.
Thus,TFθG is an oblique projection and we have(26)ranTF=TFranθG∔ranθF⊥=ranTFθG,kerTFθG=ranTGθF⊥=ranTG⊥.
The fact thatF, G are basic Ω,μ-frames implies(27)TFθG=PranTF,ranTG⊥=PℋF,ℋG⊥.
The rest of the proof is obvious.


## 4. Dilation of Oblique Dual Pairs

Gabardo and Han derived a dilation theorem ([13], Theorem 1.1]) for dual pairs of continuous frames. In this section, we consider the dilation for an oblique dual pair of basic Ω,μ-frames. Our approach relies strongly on the following result:Lemma 5.
Suppose thatF is a basic Ω,μ-frame for ℋ and that both G and E are oblique duals of F with ℋG=ℋE. If either ranθG⊂ranθE or ranθE⊂ranθG holds, then G=E.Proof.
SinceG and E are oblique duals of F, we have(28)TFθG=TFθE=I|ℋF,ℋ=ℋF∔ℋG⊥.
Note thatkerθG=ℋG⊥=ℋE⊥=kerθE, which implies θEℋ=θEℋF and θGℋ=θGℋF. Now suppose ranθG⊂ranθE. Thus θGℋF⊂θEℋF and so for any y∈ℋF, we can find z∈ℋF satisfying θEz=θGy. Therefore y=TFθGy=TFθEz=z, and thus θEx=θGx holds for all x∈ℋ. This forces that E=G, as required.The preceding lemma also implies the uniqueness of the Type II dual for a fixed direct sum decomposition ofℋ. One can see ([22], Proposition 3.3] for a similar assertion for discrete frame sequences. Now we state this result in the following:Corollary 3.
Suppose thatF is a basic Ω,μ-frame for ℋ and that V is a subspace of ℋ such that ℋ=ℋF∔V⊥. Then there exists a unique Ω,μ-frame for V which is the type II dual of F.Proof.
From Proposition1, we can find a Ω,μ-frame G for V that is a Type II dual of F. If there exists another Ω,μ-frame E for V that is also a Type II dual of F, then by Lemma 5, we get G=E.Now we state our main result of this section in the following:Theorem 1.
Suppose thatF is a basic Ω,μ-frame for ℋ and that G is an oblique dual of F. Assume both ranθF and ranθG are contained in a frame range ℳ. Then there exist a Hilbert space K⊃ℋ and a basic Ω,μ-frame F1 for K such that PF1=F, PG1=G and ranθF1=ℳ, where G1 is a (unique) Type II dual of F1 and P is the orthogonal projection from K onto ℋ.Proof.
Denote byψjj∈J an orthonormal basis for ℳ and write ℳ1:=Iℳ−QL2Ω,μ=ℳ⊖ranθG, where Q and Iℳ denote orthogonal projections from L2Ω,μ onto ranθG and ℳ respectively. By Lemma 4, we know that F0ω:=∑j∈Jψjω¯Iℳ−Qψj defines a parseval Ω,μ-frame for ℳ1. Therefore, for any φ∈ℳ1,(29)θF0φω=φ,F0ω=∑j∈JIℳ−Qφ,ψjψjω=φω,which implies ranθF0=ℳ1.
LetK=ℋ⊕ℳ1 and F1=F⊕F0. Clearly F1 is a Ω,μ-Bessel mapping for K and PF1=F, where P is the orthogonal projection from K onto ℋ. From (29) it follows that, for any x⊕φ∈K,(30)θF1x⊕φω=θFxω+θF0φω=θFxω+φω.
Then by Lemma1, we have(31)ranθF1=ranθF∔ℳ1=ℳ.
SinceθF|HF is injective and θF|HF⊥=0, it follows that θF1 is bijective from HF⊕ℳ1 onto ℳ, which means that F1 is a continuous frame for HF⊕ℳ1 with ranθF1=ℳ. Since G is an oblique dual of F, we see from Proposition 1 that ℋ=ℋF∔ℋG⊥, and thus we can write(32)K=ℋ⊕ℳ1=ℋF⊕ℳ1∔ℋG⊕ℳ1⊥.
By Corollary3, we can find a unique Type II dual G1 of F1 satisfying ℋG1=ℋG⊕ℳ1 and ranθG1=ranθF1=ℳ. Put E=PG1. Obviously, E is a continuous frame for ℋG and thus it only remains to show E=G. On the one hand, we compute, for any x∈ℋF⊂ℋF⊕ℳ1,(33)x=Px=P∫ΩPx,G1ωF1ωdμω1/2=∫Ωx,PG1ωPF1ωdμω=∫Ωx,EωFωdμω,which shows that E is a dual of F. Thus, it follows from Proposition 4 that E is an oblique dual of F. On the other hand, we compute, for any x∈ℋF,(34)x=∫ΩPx,G1ωF1ωdμω=∫Ωx,EωFωdμω⊕∫Ωx,EωF0ωdμω.
Putting (33) and (34) together, for any x∈ℋF, we have(35)∫Ωx,EωF0ωdμω=TF0θEx=0,which implies that(36)θEx∈kerTF0∩ℳ=ranθF0⊥∩ℳ=ℳ1⊥∩ℳ=ranθG,∀x∈ℋF.
Note thatθE|ℋG⊥=0 and ℋ=ℋF∔ℋG⊥, which yield that ranθE⊂ranθG. Then by Lemma 5, we get the required result.
Note that the above theorem covers Theorem 1.1 in [13].As was pointed out in the Introduction, a basic continuous frameF is a basic Riesz-type continuous frame if and only if ranθF=L2Ω,μ. Putting Lemma 2 and Theorem 1 together, we get the following result:Corollary 4.
Suppose thatΩ,μ is a purely atomic, positive measure space where μ is σ-finite, and that G is an oblique dual of F for ℋ. Then there exist a Hilbert space K⊃ℋ and a basic Riesz-type Ω,μ-frame F1 for K such that PF1=F and PG1=G, where P is the orthogonal projection from K onto ℋ and G1 is the unique dual of F1.Using Theorem1, we can investigate a dilation property for one basic continuous frame.Corollary 5.
Suppose thatF is a basic Ω,μ-frame for ℋ and that ranθF is contained in a frame range ℳ. Then there exist a Hilbert space K⊃ℋ and a basic Ω,μ-frame F1 for K such that PF1=F and ranθF1=ℳ, where P is the orthogonal projection from K onto ℋ.Proof.
From Corollary3, we can find a unique basic Ω,μ-frame G which is an oblique dual of F such that ranθF=ranθG. Then by Theorem 1, we get the required result.We can derive a dilation result for Type I duals of basicΩ,μ-frames, which is a special case of Theorem 1.Proposition 5.
Suppose thatF is a basic Ω,μ-frame for ℋ and that Gis a Type I dual ofF. Assume both ranθF and ranθG are contained in a frame range ℳ. Then, there exists a Hilbert space K⊃ℋ and a basic Ω,μ-frame F1 for K such that PF1=F, PG1=G and ranθF1=ℳ, where G1 is the canonical dual of F1 and P is the orthogonal projection from K onto ℋ.Proof.
SinceG is a Type I dual of F, it follows that ℋF=ℋG. Thus,(37)ℋF1=ℋF⊕ℳ1=ℋG⊕ℳ1=ℋG1,which means that G1 is a Type I dual of F1. Moreover, we see from Theorem 1 that G1 is a Type II dual of F1. Hence G1 is simultaneously a Type I and a Type II dual of F1, i.e., canonical dual of F1.WhenG is a Type II dual of F in Theorem 1, we cannot get a dilation result similar to the above proposition.Suppose thatF is a continuous frame and that P∈Bℋ is an orthogonal projection. It is easy to check that PF is a continuous frame for Pℋ. When P is an oblique projection, we also have the following result:Lemma 6.
Suppose thatF is a continuous frame for H and that P:=PU,V⊥∈Bℋ is an oblique projection, where U and V are closed subspaces of ℋ. Then PF is a continuous frame for U.Proof.
We compute for anyx∈ℋ,(38)θPFxω=x,PU,V⊥Fω=PV,U⊥x,Fω=θFPV,U⊥xω.Now assume θPFy=0 holds for any y∈U. Then we see from (38) that θFPV,U⊥y=0. Since θF is injective, it follows that PV,U⊥y=0, and thus y∈U⊥∩U=0. This shows that θPF|U is injective and we have the result.We have considered the dilation of oblique dual pairs under orthogonal projections in Theorem1. Now we deduce a dilation result in terms of oblique projections.Theorem 2.
Suppose thatF is a basic Ω,μ-frame for ℋ and that G is an oblique dual of F. Assume both ranθF and ranθG are contained in a frame range ℳ. If dimℳ⊖ranθG=dimℋG⊥, then there exist a Ω,μ-frame F1 for ℋ with its canonical dual G1 such that PℋF,ℋG⊥F1=F,PℋG,ℋF⊥G1=G and ranθF1=ℳ.Proof.
LetQ and Iℳ be orthogonal projections from L2Ω,μ onto ranθG and ℳ respectively and write ℳ1=Iℳ−QL2Ω,μ=ℳ⊖ranθG. Since dimℳ1=dimℋG⊥, we can find a unitary operator W∈Bℳ1,ℋG⊥. Let ψjj∈J be an orthonormal basis for ℳ and define F0=∑j∈Jψjω¯WIℳ−Qψj. For any x∈ℋG⊥, we have(39)θF0x=∑j∈JIℳ−QW−1x,ψjψj=Iℳ−QW−1x=W−1x
We see thatθF0|ℋG⊥ is injective, and thus F0 is a continuous frame for ℋG⊥ with frame range ℳ1. Put F1=F+F0. Clearly PℋF,ℋG⊥F1=F and F1 is a Ω,μ-Bessel mapping for ℋ.
Sinceℋ=ℋF⊕ℋF⊥=ℋG⊕ℋG⊥, for any y∈ℋ, we can write y=x1+z1=x2+z2∈ℋ, where x1∈ℋF, z1∈ℋF⊥, x2∈ℋG, z2∈ℋG⊥. Note that θF|ℋF⊥=0 and θF0|ℋG=0, which implies that(40)θF1y=θFx1+z1+θF0x2+z2=θFx1+θF0z2=θFx1+W−1z2.
Using Lemma1, we have ranθF1=ranθF∔ℳ1=ℳ. The fact that F is a continuous frame for ℋF shows that θF|ℋF is injective. Thus θF1 is injective, implying that F1 is a continuous frame for ℋ. Let G1 be the canonical dual of F1 and E=PℋG,ℋF⊥G1. It follows from Lemma 6 that E is a continuous frame for ℋG and thus it only remains to show E=G. For any x∈ℋF, we get(41)x=PℋF,ℋG⊥=PℋF,ℋG⊥∫Ωx,G1ωF1ωdμω=∫Ωx,PℋG,ℋF⊥G1ωPℋF,ℋG⊥F1ωdμω=∫Ωx,EωFωdμω,which implies that E is a dual of F. Then by Proposition 4, it follows that E is an oblique dual of F. For any y∈ℋF, we have(42)y=∫ΩPℋF,ℋG⊥y,G1ωF1ωdμω=∫Ωy,EωFωdμω+∫Ωy,EωF0ωdμω=y+TF0θEy.Therefore TF0θEy=0 for y∈ℋF, and thus,(43)θEy∈kerTF0∩ℳ=ranθF0⊥∩ℳ=ℳ1⊥∩ℳ=ranθG.
SinceθE|ℋG⊥=0 and ℋ=ℋF∔ℋG⊥, we have ranθE⊂ranθG and the theorem now follows by Lemma 5.


## 5. Perturbations of Basic Continuous Frames

The perturbation theorem ([13], Theorem 1.2), as a generalization of Casazza and Christensen’s result, stated a perturbation result for continuous frames. However, the spanning space of basic continuous frames may change when adding new elements. This makes the theorem in [13] no longer applicable to basic continuous frames. So we need to derive a new perturbation condition for basic continuous frames.We begin with some concepts related to subspaces. LetU, V denote closed subspaces of ℋ and define two angles between U and V:(44)SU,V≔supPVxx:x∈U∖0,ifU≠0,0,ifU=0,RU,V:=infPVxx:x∈U∖0,ifU≠0,1,ifU=0.It is known that(45)SU,V⊥=1−RU,V21/2.The following lemma relates the angle to projections.Lemma 7 (see [23]).
For closed subspacesU and V of ℋ with at least one nontrivial, the following are equivalent.(i)
0<RU,Vand0<RV,U(ii)
0<RU,V=RV,U=RU⊥,V⊥=RV⊥,U⊥(iii)
PV|U:U⟶Vis invertible(iv)
PU|V:V⟶Uis invertible(v)
ℋ=V∔V⊥(vi)
ℋ=U∔U⊥We also need the following classical fact in operator theory:Lemma 8 (see [15]).
Suppose thatL is a linear operator on a Banach space X and that there exist constants λ1,λ2∈[0,1) such that(46)Lx−x≤λ1x+λ2Lx,∀x∈ℋ.ThenL is bounded and invertible. Moreover, for all x∈X(47)1−λ11+λ2x≤Lx≤1+λ11−λ2x,1−λ21+λ1x≤L−1x≤1+λ21−λ1x.Now we give a condition for the perturbation of basic continuous frames:Theorem 3.
Suppose thatF is a basic Ω,μ-frame for ℋ with frame bounds A, B and that G:Ω⟶ℋ is a vector-valued function. If there exist constants λ1,μ≥0 and λ2∈[0,1) such that(48)∫ΩfωFω−Gωdμω≤λ1∫ΩfωFωEμω+λ2∫ΩfωGωdμω+μf,
for allf∈L2Ω,μ with μf≠0<∞, then G is a Ω,μ-Bessel mapping with a Bessel bound(49)B1+λ1+λ2+μ/B1−λ22.If furthermoreλ1+μ/A<RℋG,ℋF, then G is a basic Ω,μ-frame with a lower frame bound(50)A1−λ2+λ1+μ/A2+SℋF⊥,ℋG21+λ22.Moreover,ℋF is isomorphic to ℋG and ℋF⊥ is isomorphic to ℋG⊥.Proof.
Letℳ=f∈L2Ω,μ:μf≠0<∞ and define T:ℳ⟶ℋ by(51)Tf=∫ΩfωGωdμω.
Then, condition (48) implies that(52)TFf−Tf≤λ1TFf+λ2Tf+μf,forallf∈ℳ.
Hence,(53)Tf≤TFf+TFf−Tf≤1+λ1TFf+λ2Tf+μf≤1+λ1Bf+λ2Tf+μf,which implies that(54)Tf≤1+λ1B+μ1−λ2f.
Sinceℳ is dense in L2Ω,μ, we can extend T uniquely to a bounded operator from L2Ω,μ into ℋ. Therefore (48) holds for every f∈L2Ω,μ and G is a Ω,μ-Bessel mapping with a Bessel bound(55)1+λ1B+μ1−λ22=B1+λ1+λ2+μ/B1−λ22.
SinceG is a Ω,μ-Bessel mapping, one can define an operator L∈Bℋ by(56)Lx=∫Ωx,SF†FωGωdμω+PℋG⊥PℋF⊥x.
Note thatSF†F is the canonical dual of F with frame bounds 1/B, 1/A. Thus for any y∈ℋF,(57)y=∫Ωy,SF†FωFωdμω.
Fixx∈ℋ and write x=x1+x2, where x1∈ℋF, x2∈ℋF⊥. Using (48) and (57), we obtain(58)x−Lx≤x1−Lx1+x2−Lx2≤∫Ωx1,SF†FωFω−Gωdμω+PℋF⊥−PℋG⊥PℋF⊥x2≤λ1∫Ωx1,SF†FωFωdμω+λ2∫Ωx1,SF†FωGωdμω+μ∫Ωx1,SF†Fω2dμω1/2+I−PℋG⊥PℋF⊥x2≤λ1x1+λ2Lx1+μ/Ax1+PℋGx2≤λ1+μ/Ax1+SℋF⊥,ℋGx2+λ2Lx1≤λ1+μ/A2+SℋF⊥,ℋG2⋅x12+x22+λ2Lx=λ1+μ/A2+SℋF⊥,ℋG2⋅x+λ2Lx.If furthermore λ1+μ/A<RℋG,ℋF, then by Lemma 7 and (45),(59)λ1+μ/A2+SℋF⊥,ℋG2<RℋG,ℋF2+SℋF⊥,ℋG2=RℋF⊥,ℋG⊥2+SℋF⊥,ℋG2=1.
Thus using Lemma8, L is invertible and(60)L−1≤1+λ21−λ1+μ/A2+SℋF⊥,ℋG2.
Hence,L maps ℋF onto ℋG and ℋF⊥ onto ℋG⊥. For any x∈ℋG, it follows that L−1x∈ℋF, and thus PℋG⊥PℋF⊥L−1x=0. So, we can write(61)x=LL−1x=∫ΩL−1x,SF†FωGωdμω,which implies that(62)x4=x,x2=∫ΩL−1x,SF†FωGω,xdμω2≤∫ΩL−1x,SF†Fω2dμω⋅∫ΩGω,x2dμω≤1AL−1x2⋅∫Ωx,Gω2dμω.
Therefore,(63)∫Ωx,Gω2dμω≥A1−λ1+μ/A2+SℋF⊥,ℋG21+λ22⋅x2=A1−λ2+λ1+μ/A2+SℋF⊥,ℋG21+λ22⋅x2,as required.


## 6. Conclusion

In this paper, we investigate continuous frames for their spanning sets and call them basic continuous frames. The use of basic continuous frames allows more freedom when we design the optimal dual frame and the corresponding analysis-synthesis system. We first characterize basic continuous frames and their oblique duals by using operators with closed ranges. Then we show that an oblique dual pair of basic continuous frames for a Hilbert space can be dilated to a Type II dual pair for a larger Hilbert space. Also, we present a condition under which an oblique dual pair of basic continuous frames can be dilated to a dual pair of continuous frames for the same space. Finally, with the help of angles between different spanning subspaces, a perturbation condition for basic continuous frames is given.



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*Source: 3270281-2022-03-29.xml*