#### 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