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