Research Article Characterization, Dilation, and Perturbation of Basic Continuous Frames

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.


Introduction
e 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][10][11][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. roughout this paper, H and K denote Hilbert spaces over the complex field and J a countable index set. For an operator T ∈ B(H, K), we denote its range and kernel by ran T and ker T, 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 ∈ B(H) a (oblique) projection if it satisfies P 2 � P. In this case, the decomposition H � ran P∔ker P holds. Conversely, if H � U∔V, then we can find a unique projection P U,V satisfying ran P U,V � U and ker P U,V � V. Let P U : � P U,U ⊥ be the orthogonal projection onto U and P U | V be its restriction to V.
Let (Ω, 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, F(ω)〉 is a measurable function on Ω. (ii) ere exist constants A, B > 0 such that A‖x‖ 2 ≤ Ω |〈x, F(ω)〉| 2 dμ(ω) ≤ B‖x‖ 2 , ∀x ∈ H. (1) e numbers A, 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. e 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 space span F(ω) { } ω∈Ω . When Ω � N and μ is the counting measure, the family F(ω) { } ω∈Ω is a discrete frame. To a (Ω, μ)-Bessel mapping F, we associate the analysis operator θ F given by Note that the frame condition (1) ‖x‖. e adjoint T F of θ F is called synthesis operator and is given by In this case, we usually say that the following equation holds in the weak sense: e frame operator S F is defined to be T F θ F . When F is a continuous frame for H, S F is a bounded, invertible and positive operator.
Every (Ω, μ)-frame always has a dual S −1 F F, called the canonical dual, satisfying the following reconstruction formula: If F has the unique dual, then we say F is a Riesz-type (Ω, 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 L 2 (Ω, 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. e paper is organized as follows. In Section 2, 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.

Preliminaries
is 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: For a given vector-valued function F: Ω ⟶ H. For a (Ω, μ)-Bessel mapping F, we always use θ F and T F to denote the analysis operator and synthesis operator, respectively.
Recall that every basic continuous frame is a (Ω, μ)-Bessel mapping. Conversely, a (Ω, μ)-Bessel mapping F constitutes a basic continuous frame if and only if ran T F is closed. In this case, For a basic (Ω, μ)-frame F, the frame operator S F is invertible when restricted to span F(ω) [17]). We recall that Now, we give definitions on duals of basic (Ω, μ)-frames with the following. For the discrete case, we see [18][19][20] Definition 1. Suppose that F is a basic (Ω, μ)-frame and that G is a (Ω, μ)-Bessel mapping for H.
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. ey 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 ran T G � ran T F ; if G is a Type II dual of F, then G is an oblique dual of F and ran θ G � ran θ F . Similar to ([20], eorem 1.4), we also have: Suppose that U and V are closed subspaces of H and that F is a (Ω, μ)-frame for U. en the following are equivalent: From the above proposition, the fact that G is an oblique dual of F shows H � H F ∔H ⊥ G . For a pair of oblique duals, L 2 (Ω, μ) also have a decomposition.

Lemma 1.
Suppose that F is a basic (Ω, μ)-frame for H and that G is an oblique dual of F. Assume M ⊂ L 2 (Ω, μ) is closed containing both ran θ F and ran θ G . en, (1) By assumption, we have Hence, which implies that θ F T G is a projection. Using (9), we obtain erefore, ker(θ F T G ) � (ran(θ G T F )) ⊥ � (ranθ G ) ⊥ and thus θ F T G � P ran θ F ,ran θ ⊥ G . is implies that (ii). Note that (i) shows that ranθ F ∩ (M⊖ranθ G ) � 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. ey showed that not all closed subspaces of L 2 (Ω, μ) constitute frame ranges: Lemma 2 (see [21]). Let (Ω, μ) be a σ-finite, positive measure space.
purely atomic (ii) Every closed subspace of a frame range is a frame range We finish this section with the following equivalent condition about the frame range: Lemma 3 (see [13], Corollary 2.9). For a closed subspace M of L 2 (Ω, μ), the following are equivalent:

Characterization of Basic Continuous Frames and Their Oblique Duals
is section focuses on the characterization of basic continuous frames and their oblique duals. We begin with a characterization for basic parseval (Ω, μ)-frames.

Lemma 4.
Suppose that (Ω, μ) is a measure space with positive measure μ. en the following are equivalent: whose range is a frame range and some orthonormal basis ψ j j∈J of ran θ (iii) F(ω) � j∈J ψ j (ω)e j for some orthonormal set e j j∈J of H and some orthonormal basis ψ j j∈J of a frame range M Proof. (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 ortho- . Write M � ranθ and e j � θ * ψ j for every j ∈ J. Since θ is a partial isometry, θ * is an isometry restricted to (kerθ * ) ⊥ � ranθ � M. Hence e j j∈J is an orthonormal set.
implying that F is a basic parseval continuous frame. □ Similarly, we can deduce the following characterization for basic (Ω, μ)-frames.

Proposition 2.
Suppose that (Ω, μ) is a measure space with positive measure μ. en the following are equivalent: θ: H ⟶ L 2 (Ω, μ) whose range is a frame range and some orthonormal basis of ranθ (iii) F(ω) � j∈J ψ j (ω)e j for some Riesz sequence e j j∈J of H and some orthonormal basis ψ j j∈J of a frame range M e implication (iii) ⇒ (i) now follows from the fact that e j j∈J is a Riesz sequence. □ Suppose now M is a frame range and that ψ j j∈J is an orthonormal basis for M. We compute implying that ω ⟶ j∈J ψ j (ω)ψ j 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 where θ * is invertible restricted to M ⟶ ranθ * . is means that every basic (Ω, μ)-frame is similar to a basic parseval (Ω, μ)-frame for L 2 (Ω, μ). Putting Lemma 2 and Proposition 2 together, we can characterize basic Riesz-type (Ω, μ)-frames: □ Corollary 1. Suppose that (Ω, μ) is a purely atomic, positive measure space where μ is σ-finite, and that ψ j j∈J is an orthonormal basis for L 2 (Ω, μ). en, the following are equivalent: It is known that there is a bijective correspondence between the set of discrete frame sequences and all the operators with closed range. e following proposition derives a corresponding result for basic continuous frames.
is is the essential difference between discrete frame sequences and basic continuous frames.

Proposition 3.
Every basic (Ω, μ)-frame corresponds to an operator whose range is a frame range.
Proof. Suppose θ: H ⟶ L 2 (Ω, μ) is an operator whose range is a frame range. en, by Proposition 2, F(ω) � j∈J ψ j (ω)θ * ψ j is a basic continuous frame for H, where ψ j∈J is an orthonormal basis for ranθ. Moreover, we compute which implies that θ is the analysis operator for F.

□
A new basic (Ω, 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. Proof. By assumption, WF is a (Ω, μ)-Bessel mapping with associated analysis operator θ F W * . Now, suppose WF is a basic continuous frame. en, the analysis operator θ F W * has a closed range and so is Wθ * F . is means that W has a closed range restricted to ranθ * F � span F(ω) { } ω∈Ω . Conversely, suppose W has a closed range. Since ran(θ F W * ) ⊂ ranθ F and ranθ F is a frame range, we see from Lemma 2 (ii) that ran(θ F W * ) is also a frame range. en, by Proposition 3, WF is a basic continuous frame.

□
We finish this section with the following characterization for an oblique dual pair of basic (Ω, μ)-frames.
. By the definition of oblique duals, we get that Hence, we have which implies that T F θ G is an oblique projection with ran(T F θ G ) � H F and ker( Using Cauchy Schwarz' inequality and that G is a (Ω, μ)-Bessel mapping, we can deduce the lower frame condition for F. erefore F is a (Ω, μ)-frame for H F 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). e existence of P H F ,H ⊥ G implies that Hence (i) ⇒ (vi) follows from the proof of Lemma 1.
From the identity it follows that us, T F θ G is an oblique projection and we have e fact that F, G are basic (Ω, μ)-frames implies e rest of the proof is obvious. Proof. Since G and E are oblique duals of F, we have

Dilation of Oblique Dual Pairs
(28)  F is a basic (Ω, μ)-frame for H and that G is an oblique dual of F. Assume both ranθ F and ranθ G are contained in a frame range M. en there exist a Hilbert space K H and a basic (Ω, μ)-frame F 1 for K such that PF 1 � F, PG 1 � G and ranθ F 1 � M, where G 1 is a (unique) Type II dual of F 1 and P is the orthogonal projection from K onto H.
Let K � H ⊕ M 1 and F 1 � F ⊕ F 0 . Clearly F 1 is a (Ω, μ)-Bessel mapping for K and PF 1 � F, where P is the orthogonal projection from K onto H. From (29) it follows that, for any x ⊕ φ ∈ K, en by Lemma 1, we have Since θ F | H F is injective and θ F | H ⊥ F � 0, it follows that θ F 1 is bijective from H F ⊕ M 1 onto M, which means that F 1 is a continuous frame for H F ⊕ M 1 with ranθ F 1 � M. Since G is an oblique dual of F, we see from Proposition 1 that H � H F ∔H ⊥ G , and thus we can write By Corollary 3, we can find a unique Type II dual G 1 of Obviously, E is a continuous frame for H G and thus it only remains to show E � G. On the one hand, we compute, for any x ∈ H F ⊂ H F ⊕ M 1 , which shows that E is a dual of F. us, it follows from Proposition 4 that E is an oblique dual of F. On the other hand, we compute, for any x ∈ H F , (34)

Putting (33) and (34) together, for any
which implies that Note that θ E | H ⊥ G � 0 and H � H F ∔H ⊥ G , which yield that ranθ E ⊂ ranθ G . en by Lemma 5, we get the required result.
Note that the above theorem covers eorem 1.1 in [13].
□ As was pointed out in the Introduction, a basic continuous frame F is a basic Riesz-type continuous frame if and only if ranθ F � L 2 (Ω, μ). Putting Lemma 2 and eorem 1 together, we get the following result: Corollary 4. Suppose that (Ω, μ) is a purely atomic, positive measure space where μ is σ-finite, and that G is an oblique dual of F for H. en there exist a Hilbert space K H and a basic Riesz-type (Ω, μ)-frame F 1 for K such that PF 1 � F and PG 1 � G, where P is the orthogonal projection from K onto H and G 1 is the unique dual of F 1 .
Using eorem 1, we can investigate a dilation property for one basic continuous frame.

Corollary 5.
Suppose that F is a basic (Ω, μ)-frame for H and that ranθ F is contained in a frame range M. en there exist a Hilbert space K H and a basic (Ω, μ)-frame F 1 for K such that PF 1 � F and ranθ F 1 � M, where P is the orthogonal projection from K onto H.
Proof. From Corollary 3, we can find a unique basic (Ω, μ)-frame G which is an oblique dual of F such that ranθ F � ranθ G . en by eorem 1, we get the required result.

□
We can derive a dilation result for Type I duals of basic (Ω, μ)-frames, which is a special case of eorem 1. F is a basic (Ω, μ)-frame for H and that G is a Type I dual of F. Assume both ranθ F and ranθ G are contained in a frame range M. en, there exists a Hilbert space K H and a basic (Ω, μ)-frame F 1 for K such that PF 1 � F, PG 1 � G and ranθ F 1 � M, where G 1 is the canonical dual of F 1 and P is the orthogonal projection from K onto H.

Proposition 5. Suppose that
Proof. Since G is a Type I dual of F, it follows that H F � H G . us, which means that G 1 is a Type I dual of F 1 . Moreover, we see from eorem 1 that G 1 is a Type II dual of F 1 . Hence G 1 is simultaneously a Type I and a Type II dual of F 1 , i.e., canonical dual of F 1 .

□
When G is a Type II dual of F in eorem 1, we cannot get a dilation result similar to the above proposition.
Suppose that F is a continuous frame and that P ∈ B(H) 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 that F is a continuous frame for H and that P: � P U,V ⊥ ∈ B(H) is an oblique projection, where U and V are closed subspaces of H. en PF is a continuous frame for U.
Proof. We compute for any x ∈ H, Now assume θ PF y � 0 holds for any y ∈ U. en we see from (38) that θ F P V,U ⊥ y � 0. Since θ F is injective, it follows that P V,U ⊥ y � 0, and thus y ∈ U ⊥ ∩ U � 0 { }. is shows that θ PF | U is injective and we have the result.

□
We have considered the dilation of oblique dual pairs under orthogonal projections in eorem 1. Now we deduce a dilation result in terms of oblique projections.

Theorem 2.
Suppose that F is a basic (Ω, μ)-frame for H and that G is an oblique dual of F. Assume both ranθ F and ranθ G are contained in a frame range M. If dim(M⊖ranθ G ) � dimH ⊥ G , then there exist a (Ω, μ)-frame Proof. Let Q and I M be orthogonal projections from L 2 (Ω, μ) onto ranθ G and M respectively and write . Let ψ j j∈J be an orthonormal basis for M and define We see that θ F 0 | H ⊥ G is injective, and thus F 0 is a continuous frame for H ⊥ G with frame range M 1 . Put (40) Using Lemma 1, we have ranθ F 1 � ranθ F ∔M 1 � M. e fact that F is a continuous frame for H F shows that θ F | H F is injective. us θ F 1 is injective, implying that F 1 is a continuous frame for H. Let G 1 be the canonical dual of F 1 and E � P H G ,H ⊥ F G 1 . It follows from Lemma 6 that E is a continuous frame for H G and thus it only remains to show E � G. For any x ∈ H F , we get which implies that E is a dual of F. en by Proposition 4, it follows that E is an oblique dual of F. For any y ∈ H F , we have erefore T F 0 θ E y � 0 for y ∈ H F , and thus, Since θ E | H ⊥ G � 0 and H � H F ∔H ⊥ G , we have ran θ E ⊂ ranθ G and the theorem now follows by Lemma 5. □

Perturbations of Basic Continuous Frames
e perturbation theorem ( [13], eorem 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. is 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. Let U, V denote closed subspaces of H and define two angles between U and V:

Journal of Mathematics
It is known that e following lemma relates the angle to projections.
Lemma 7 (see [23]). For closed subspaces U and V of H with at least one nontrivial, the following are equivalent.
We also need the following classical fact in operator theory: Lemma 8 (see [15]). Suppose that L is a linear operator on a Banach space X and that there exist constants λ 1 , λ 2 ∈ [0, 1) such that ‖Lx − x‖ ≤ λ 1 ‖x‖ + λ 2 ‖Lx‖, ∀x ∈ H. (46) en L is bounded and invertible. Moreover, for all Now we give a condition for the perturbation of basic continuous frames: Suppose that F is a basic (Ω, μ)-frame for H with frame bounds A, B and that G: Ω ⟶ H is a vectorvalued function. If there exist constants λ 1 , μ ≥ 0 and for all f ∈ L 2 (Ω, μ) with μ( f ≠ 0 ) < ∞, then G is a (Ω, μ)-Bessel mapping with a Bessel bound (49) H F ), then G is a basic (Ω, μ)-frame with a lower frame bound en, condition (48) implies that Hence, which implies that Since M is dense in L 2 (Ω, μ), we can extend T uniquely to a bounded operator from L 2 (Ω, μ) into H. erefore (48) holds for every f ∈ L 2 (Ω, μ) and G is a (Ω, μ)-Bessel mapping with a Bessel bound (55) Since G is a (Ω, μ)-Bessel mapping, one can define an operator L ∈ B(H) by Note that S † F F is the canonical dual of F with frame bounds 1/B, 1/A. us for any y ∈ H F , If furthermore λ 1 + μ/ � � A √ < R(H G , H F ), then by Lemma 7 and (45), us using Lemma 8, L is invertible and Hence, L maps H F onto H G and H ⊥ F onto H ⊥ G . For any x ∈ H G , it follows that L − 1 x ∈ H F , and thus P H ⊥ G P H ⊥ F L − 1 x � 0. So, we can write which implies that erefore, as required.
□ Journal of Mathematics

Conclusion
In this paper, we investigate continuous frames for their spanning sets and call them basic continuous frames. e 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. en 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.