New Properties of Dual Continuous -g-Frames in Hilbert Spaces

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 continuous -g-frames in Hilbert spaces. Also, we establish some new properties.

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].

Let be a Banach space, a measure space, and function a measurable function. The integral of the Banach-valued function was defined by Bochner and others. Most properties of this integral are similar to those of the integral of real-valued functions.

Let be a measure space, let and be two separable Hilbert spaces, is a sequence of subspaces of , we denote as the collection of all bounded linear operators from to , and is abbreviated to .

For , we use the notation and to denote, respectively, the range and the null space of .

Definition 1 (see [4]). A sequence is called a continuous g-frame for with respect to if there exists such thatThe numbers and are called lower and upper bounds of the continuous g-frame, respectively.
If , the frame is -tight.
If , it is called a normalized tight continuous g-frame or a Parseval continuous g-frame.
We call a continuous g-Bessel sequence if the right-hand inequality of (1) holds.

Let be a continuous g-frame for with respect to .

The synthesis operator of is defined by , given bywhere .

The existence of implies that is well defined and bounded.

The adjoint operator of which is given byis said to be the analysis operator of .

Definition 2 (see [5]). Let . A sequence is called a continuous K-g-frame for with respect to if there exists such thatThe numbers and are called lower and upper bounds of the continuous K-g-frame, respectively.
If , the sequence is called a continuous K-g-frame for with respect to .
The -tight continuous K-g-frame for is said to be a Parseval continuous K-g-frame if .
Suppose that and is a Parseval continuous K-g-frame for with respect to with synthesis operator . Then, it is easy to check

Definition 3. Let and be a continuous K-g-frame for with respect to . A continuous K-g-Bessel sequence for with respect to is said to be a dual continuous K-g-Bessel sequence of if

Remark 1. (1)If , then a dual continuous K-g-Bessel sequence is just a dual continuous g-frame(2)A dual continuous K-g-Bessel sequence is necessarily a continuous -g-frame(3)Denote the synthesis operator of and by and , respectively; then, (6) means that

Lemma 1 (see [6]). Suppose that has a closed range; then, there exists a unique operator , called the pseudo-inverse of , satisfying

Lemma 2 (see [7]). Let , , and be three Hilbert spaces; also, let and . The following statements are equivalent:(1)(2)There exists such that (3)There exists such that Moreover, if (1)–(3) are valid, then there exists a unique operator such that(a)(b)(c)

In the following section, we set out to generalize some results of Xiang [8]; in other words, we characterize the concept of dual continuous -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 that has a closed range, and let to be a continuous K-g-frame for with respect to and be a dual continuous K-g-Bessel sequence of ; then, is a continuous K-g-frame for with respect to .

Proof. For each , we have , and thenwhich givewhere and are the upper bounds of and , respectively.
Consequently,Assume that has a closed range; hence, for each , we get , by Lemma 1, and thus,Now,which gives is a continuous K-g-frame for with respect to with bounds and .

Proposition 2. Let ; then, every continuous K-g-frame admits a dual continuous K-g-Bessel sequence.

Proof. Let be a continuous K-g-frame for with respect to , with bounds and ; then, for each , we havewhich is equivalent to ; by Lemma 2, there exists such that .
Let be the projection on that maps each element to its -th component, i.e., , where if , and if not.
If we let , thenHence, is a continuous g-Bessel sequence for with respect to . Now,showing that is a continuous g-Bessel sequence.

Proposition 3. Suppose that and is a continuous K-g-frame for with respect to ; then, there exists a unique dual continuous K-Bessel sequence of such that for any dual continuous K-g-Bessel sequence of .

Proof. Since , by Lemma 2, it follows that there is a unique operator such that andtaking for each ; then, as shown in Proposition 2, is a continuous g-Bessel sequence for with respect to , and furthermore, it is a dual continuous K-g-Bessel sequence of .
Since for any and , .
Now, let be any dual continuous K-g-Bessel sequence of ; then, , and as a sequence. For each , we haveIt follows that .
Equivalently, .

Proposition 4. Suppose that and is a continuous K-g-frame for with respect to ; then, the dual continuous K-g-Bessel sequence of is precisely the family , where satisfies the condition .

Theorem 1. Let and be a Parseval continuous K-g-frame for with respect to , where has a closed range; then, is a dual continuous K-g-Bessel sequence of .

Proof. It is easy to see that is a continuous K-g-frame for with respect to .
By Lemma 1, for every ; thus, by (5),If , then by Lemma 1 again, we obtain , implying that . Altogether, we havewhich ends the proof.

Theorem 2. Let have a closed range and be a Parseval continuous K-g-frame for with respect to . For each , let ; then, is a dual continuous K-g-Bessel sequence of for upper bound if and only if there exists such that and for each . In this case, the continuous g-Bessel bound of is .

Proof. First, assume that is a dual continuous K-g-Bessel sequence of , and we define by for each and ; then, . Indeed,for each , and by Theorem 1, we haveConversely, we show that is a continuous g-Bessel sequence for with respect to with bound .
From Theorem 1, we have

Corollary 1. Let have a closed range and be a Parseval continuous K-g-frame for with respect to , and let ; then, is a dual continuous K-g-Bessel sequence of if and only if there exists a continuous g-Bessel sequence for with respect to such that

Proof. Assume that is a dual continuous K-g-Bessel sequence of ; then, from Theorem 2, there exists such thatTaking , then clearly, is a continuous g-Bessel sequence for with respect to with bound , andConversely, let , where is a continuous g-Bessel sequence for with respect to such thatBy Theorem 1,

Theorem 3. Let have a closed range and be a Parseval continuous K-g-frame for with respect to ; then, is the canonical dual continuous K-g-Bessel sequence of .

Proof. By Theorem 1, we know that is a dual continuous K-g-Bessel sequence of ; to complete this proof, we need to prove by Proposition 3 that for any dual continuous K-g-Bessel sequence of , where is the synthesis operator of . By Theorem 2, there exists such that and . A simple computation gives , noting .
For each , we haveTherefore, .

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 operator T on a Banach space is invertible if we have .

Lemma 3 (see [9]; Paley–Wiener). Let be a basis of a Banach space and a sequence of vectors in . If there exists a constant such thatfor all finite sequences of scalars, then is also a basis for .

Proposition 5. Let have a closed range, be a continuous K-g-frame for with respect to with bounds , and be a dual continuous K-g-Bessel sequence of with upper bound . For each , let ; if the following conditions are satisfied,(1) and(2),then is a continuous K-g-frame for with respect to with continuous K-g-frame bounds and , where is defined by such that and denotes the orthogonal projection on .

Proof. DefineThen, (1) implies that is well defined and bounded with .
Clearly, is linear.
Thus,Therefore, is a continuous K-g-Bessel sequence for with respect to .
For any , we haveHence,From this, we conclude that the operator is invertible with . It is trivial to show that is closed.
For any , we haveTherefore,It follows thatThis completes the proof.

3. Continuous K-g-Frame Sequences and Continuous g-Frame Sequences

Theorem 4. Let be a g-frame for with respect to ; then, is a continuous K-g-frame for with respect to if and only if .

Proof. Suppose that is a continuous K-g-frame for with respect to ; then, , we haveFrom Lemma 2, we have since . Conversely, let .
Since , we haveFor any .
For all , , soOn the contrary, since , ; then, ; hence, .
Then,

Proposition 6. Let have a closed range and be a continuous K-g-frame for with respect to ; then, is a continuous g-frame for with respect to .

Proof. We just prove the lower continuous g-frame inequality. From Proposition 2, there exists a continuous g-Bessel sequence for with respect to such that .
For every , by Lemma 1,Take for every .
Then, we haveSo, is a continuous g-Bessel sequence for with respect to .
For any , we haveand then .
Hence,Then, is a continuous g-frame for with respect to .

Proposition 7. Let have a closed range and be a continuous K-g-frame for with respect to . If , then is a continuous K-g-frame for with respect to .

Proof. For each , we have such that and . So, .
Since and ,Let be a dual continuous g-frame of .
So,Hence,Then, .
On the contrary, we have .
Then, .
So, is a continuous K-g-frame for .

Theorem 5. For every , let ; then, the following statements are equivalent:(1) is a continuous K-g-frame sequence for with respect to .(2) is a continuous g-Bessel sequence for with respect to , and there exists a continuous g-Bessel sequence for with respect to such that(3) is a continuous g-Bessel sequence for with respect to , and there exists a continuous g-Bessel sequence for with respect to such that

Proof.  : We have then their adjoint By the definition on the continuous K-g-frame sequence, there exists such that This implies From Lemma 2, there exists such that For every , denote ; then, we have Hence, is a continuous g-Bessel sequence for with respect to . On the contrary, we have for all , : For every and , by (48), we have So, Hence, . :Suppose that (49) holds; to prove (1), we just prove the lower bound inequality on continuous K-g-frames.
For every , we haveSo, .

Theorem 6. A sequence is a continuous K-g-frame sequence for with respect to if and only if there exists such that

Proof. Let be a continuous K-g-frame sequence for with respect to , and let .
Hence, for every and any . So, .
From (53), we haveBy Lemma 2, it follows that .
Conversely, for any .
Let ; we haveSo, is a continuous K-g-frame sequence for with respect to .
Since , by Lemma 2, there exists such thatHence, for every , we haveand then is a continuous K-g-frame sequence for with respect to with bounds and .

In general, a continuous K-g-frame for with respect to is not a continuous g-frame for with respect to .

Proposition 8. Let be a continuous K-g-frame sequence for with respect to . Suppose that and that it has a closed range; then, is a continuous g-frame for with respect to .

Proof. Since has a closed range, its pseudo-inverse exists; from Lemma 1, every can be written asHence,Noting that , we haveSo,Since , its pseudo-inverseand thenOn the contrary, we haveSo, is a continuous g-frame for with respect to with bounds and .

In the following, we give a necessary and sufficient condition under which a tight continuous K-g-frame sequence for with respect to is a tight continuous g-frame for with respect to .

Proof. Theorem 7Let be a -tight continuous g-frame for with respect to ; then, is a -tight continuous g-frame for with respect to if and only if admits a left inverse .

Proof. First, we suppose that is a -tight continuous g-frame for with respect to .
Hence,So, for every . Then,This implies thatHence, is a left inverse of ; conversely, we haveSo,It follows that is a -tight continuous g-frame for with respect to .

Theorem 7. Let and be a -tight continuous -g-frame sequence for with respect to ; then, is a -tight continuous -g-frame sequence for with respect to if and only if

Proof. First, we suppose that is a -tight continuous -g-frame sequence for with respect to ; then, there exists such thatWe have is a -tight continuous -g-frame sequence for with respect to ; then, for every .
Hence,Therefore,By Lemma 2, we haveConversely, suppose .
By Lemma 2, there exists such thatso for every , we haveThen, for all ,This completes the proof.

4. Conclusion

In this article, we have proved some properties of dual continuous -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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.