#### Abstract

We introduce the modular continuous g-Riesz basis to improve one existing result for continuous g-Riesz basis in Hilbert -modules, and then we study the equivalency relations between continuous g-frames in Hilbert -modules, and, in particular, we obtain two necessary and sufficient conditions under which two continuous g-frames are similar. Finally, we generalize a stability result for alternate duals of g-frames in Hilbert spaces to alternate duals of continuous g-frames in Hilbert -modules.

#### 1. Introduction

Frames for Hilbert spaces were first introduced by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series, reintroduced in 1986 by Daubechies et al. [2] and popularized from then on. The theory of frames plays an important role in theoretics and applications, which has been extensively applied in signal processing, sampling theory, system modelling, and many other fields. We refer to [3–9] for an introduction to frame theory and its applications.

The theory of frames was rapidly generalized and, until 2006, various generalizations consisting of vectors in Hilbert spaces were developed. In 2006, Sun introduced the concept of g-frame in a Hilbert space in [10] and showed that this includes more of the other cases of generalizations of frame concept and proved that many basic properties can be derived within this more general context.

On the other hand, the concept of frames especially the g-frames was introduced in Hilbert -modules, and some of their properties were investigated in [11–13]. As for Hilbert -module, it is a generalization of Hilbert spaces by allowing the inner product to take values in a -algebra rather than the field of complex numbers. Note that the theory of Hilbert -modules is quite different from that of Hilbert spaces. Unlike Hilbert space cases, not every closed submodule of a Hilbert -module is complemented. Moreover, the well-known Riesz representation theorem for continuous functionals in Hilbert spaces does not hold in Hilbert -modules, which implies that not all bounded linear operators on Hilbert -modules are adjointable. It should also be remarked that, due to the complexity of the -algebras involved in the Hilbert -modules and the fact that some useful techniques available in Hilbert spaces are either absent or unknown in Hilbert -modules, the problems about frames and g-frames for Hilbert -modules are more complicated than those for Hilbert spaces. This makes the study of the frames for Hilbert -modules important and interesting. The properties of g-frames for Hilbert -modules were further investigated in [14, 15].

The concept of a generalization of frames to a family indexed by some locally compact space endowed with a Radon measure was proposed by Kaiser [16] and independently by Ali et al. [17]. These frames are known as continuous frames. Gabardo and Han in [18] called these frames “Frames associated with measurable spaces”; Askari-Hemmat et al. in [19] called them generalized frames, and in mathematical physics they are referred to as coherent states [20].

The continuous g-frames in Hilbert -modules, which were proposed by Kouchi and Nazari in [21], are an extension to g-frames in Hilbert -modules and continuous frames in Hilbert spaces, and they made a discussion of some properties of continuous g-frames in Hilbert -modules in some aspects. The purpose of this paper is to further investigate the properties of continuous g-frames in Hilbert -modules.

The paper is organized in the following manner. We continue this introductory section with a review of the basic definitions and notations about Hilbert -modules. Section 2 investigates some basic results of continuous g-frames in Hilbert -modules and introduces the so-called modular continuous g-Riesz basis to improve one result for continuous g-Riesz basis obtained by Kouchi and Nazari plus a bit more. Equivalency relations between continuous g-frames are included in Section 3, where two necessary and sufficient conditions for two continuous g-frames to be similar are obtained. The last section of this paper generalizes a stability result for alternate duals of g-frames in Hilbert spaces to alternate duals of continuous g-frames in Hilbert -modules.

Let us recall the definitions and some basic properties of Hilbert -modules. For more details, the interested readers can refer to the books by Lance [22] and Wegge-Olsen [23]. Let be a -algebra with involution . A pre-Hilbert -module over or, simply, a pre-Hilbert -module, is a complex linear space which is a left -module with map , called an -valued inner product, and it possesses the following properties:(1) for all and if and only if ;(2) for all ;(3) for all , ;(4) whenever and .

For , we define a norm on by . If is complete with this norm, it is called a Hilbert -module over or a Hilbert -module.

Let and be two Hilbert -modules. A map is said to be adjointable if there exists a map such that for all and . We denote by the collection of all adjointable -linear maps from to . The following two lemmas will be used in the later section.

Lemma 1 (see [24]). *Let and be two Hilbert -modules over a -algebra and let be a linear map. Then the following conditions are equivalent:*(1)*the operator is bounded and -linear;*(2)*there exists a constant such that the inequality holds in for all .*

Lemma 2 (see [25]). *Let be a -algebra, let and be two Hilbert -modules, and let . The following statements are equivalent:*(1)* is surjective;*(2)* is bounded below with respect to norm; that is, there is such that for all ;*(3)* is bounded below with respect to inner product; that is, there is such that for all . *

Let be a Hilbert -module and a sequence of closed submodules of . Set

For any and , if the -valued inner product is defined by and the norm is defined by , then is a Hilbert -module (see [22]).

Throughout this paper, is a unital -algebra, and are Hilbert -modules, and is a sequence of closed submodules of . For , we use and to denote the range and the null space of , respectively. As usual, we use to denote the identity operator on .

#### 2. Basic Results of Continuous g-Frames and Modular Continuous g-Riesz Bases

In this section, we recall some basic properties of continuous g-frames in Hilbert -modules and, in particular, we obtain an equivalent condition under which a Hilbert -module has a continuous g-frame. Moreover, we introduce the modular continuous g-Riesz basis to improve one result for continuous g-Riesz basis in Hilbert -modules.

*Definition 3 (see [21]). *We call a family of adjointable -linear operators a continuous generalized frame or simply a continuous g-frame for Hilbert -module with respect to if(1)for any , the function defined by is measurable;(2)there is a pair of constants such that, for any ,
The constants and are called continuous g-frame bounds. We call a continuous tight g-frame if and a continuous Parseval g-frame if . If only the right-hand inequality of (2) is satisfied, we call a continuous g-Bessel sequence for with respect to with Bessel bound .

We have the following equivalent definition for continuous g-Bessel sequences in Hilbert -modules.

Proposition 4. *Let be a sequence of adjointable -linear operators on . Then is a continuous g-Bessel sequence with Bessel bound if and only if, for all ,
*

*Proof. *“⇒”. It is obvious.

“⇐”. Define a linear operator by for all . Then
which implies that . Hence, is bounded. It is clear that is -linear. Then by Lemma 1, we have , equivalently, , as desired.

The following proposition gives an equivalent condition for a continuous g-Bessel sequence to be a continuous g-frame.

Proposition 5. *Let be a continuous g-Bessel sequence for with respect to . Then is a continuous g-frame for if and only if there exists a constant such that
*

*Proof. *“⇒”. It is straightforward.

“⇐”. We define a linear operator as follows:
Then is adjointable. Indeed,
for all . It follows from (5) that
Thus, for all . Then by Lemma 2, there exists such that ; that is, . The proof is over.

Using the above equivalent definition of continuous g-frames we can easily prove the following result that will be used in the proof of Lemma 20.

Proposition 6. *Let and be two continuous g-Bessel sequences for with respect to . If holds for all , then both and are continuous g-frames for with respect to . *

*Proof. *Let us denote the Bessel bound of by . For all , we have
It follows that
Similarly, we can show that is a continuous g-frame for .

Let be a continuous g-Bessel sequence for with respect to , we define the synthesis operator by It follows immediately from the observation that for all , and is adjointable and its adjoint operator is given by for all . We call the analysis operator. By composing and , we obtain the frame operator . Note that is a positive, self-adjoint operator which is invertible if and only if is a continuous g-frame of . If is a continuous g-frame, then every has a representation of the form

We can characterize the continuous g-frames in Hilbert -modules in terms of the associated synthesis and analysis operators.

Proposition 7. *Let be a family of adjointable -linear operators on . Then the following statements are equivalent:*(1)* is a continuous g-frame for ** with respect to **;*(2)*the synthesis operator ** is well defined and surjective;*(3)*the analysis operator ** is bounded below with respect to norm.*

*Proof. *. See [21, Theorem 4.3].

. It follows directly from Lemma 2.

We are now ready to present a necessary and sufficient condition for a Hilbert -module to have a continuous g-frame.

Theorem 8. *A Hilbert -module has a continuous g-frame with respect to if and only if there exists an adjointable and invertible map from to a closed submodule of .*

*Proof. *“⇒”. Assume that is a continuous g-frame for with respect to with synthesis operator . It follows from Proposition 7 that the analysis operator is bounded below with respect to norm; and, consequently, is injective with closed range. Now, is an adjointable and invertible map from to , which is a closed submodule of .

“⇐”. Suppose that is a closed submodule of and is an adjointable and invertible map. We define a family of adjointable operators as follows:
Taking for each , then
Hence, by [22, Proposition 1.2], we have

It is easy to see that a continuous g-Bessel sequence for with respect to is a continuous g-frame if and only if there exists a continuous g-Bessel sequence for with respect to such that In this case, we call a dual continuous g-frame of . If is the frame operator of , a continuous g-frame for with respect to , then, a direct calculation yields that is a dual continuous g-frame of ; it is called the canonical dual. A dual which is not the canonical dual is called an alternate dual or simply a dual.

Our next result is a generalization of Lemma 2.1 in [10].

Proposition 9. *Let be a continuous g-frame for with respect to which possesses more than one dual, and let be the frame operator for . Then for any dual continuous g-frame of , the inequality
**
is valid for all . Besides, the quality holds precisely if for all .**More generally, whenever for certain , we have
*

*Proof. *We begin with showing the first statement. Since is a dual continuous g-frame of , it follows that for all . Therefore,
showing that the first part of the assertion holds since

Now, suppose that has two decompositions
Since
it follows that

*Definition 10 (see [21]). *A continuous g-frame for Hilbert -module with respect to is said to be a continuous g-Riesz basis if it satisfies the following:(1) for any ;(2)if , then is equal to zero for each , where and is a measurable subset of .

By using the synthesis operator, Kouchi and Nazari gave a characterization for continuous g-Riesz basis as follows.

Theorem 11 (see [21]). *A family of adjointable -linear operators is a continuous g-Riesz basis for with respect to if and only if the synthesis operator is a homeomorphism. *

We note, however, that in the proof of the above theorem, they said that “ for any and , so ”, which is not true, because if has a dense range, then is one-to-one. We can improve their result by introducing the following modular continuous g-Riesz basis.

*Definition 12 (see [26]). *We call a family of adjointable -linear operators on a modular continuous g-Riesz basis if(1);(2)there exist constants such that for any ,

Theorem 13 (see [26]). *A sequence is a modular continuous g-Riesz basis for with respect to if and only if the synthesis operator is a homeomorphism. *

*Proof. *Suppose first that is a modular continuous g-Riesz basis for with synthesis operator . Then (25) turns to be
showing that is bounded below with respect to norm. Hence, by Lemma 2, its adjoint operator is surjective. Since condition (1) in Definition 12 implies that is injective, it follows that is invertible, and so is invertible.

Conversely, let be a homeomorphism. Then is injective. So condition (1) in Definition 12 holds. Now, for any ,
Therefore, is a modular continuous g-Riesz basis for with respect to .

The following is an immediate consequence of Theorem 13.

Corollary 14. *Let be a continuous g-frame for with respect to with synthesis operator , then it is a modular continuous g-Riesz basis for with respect to if and only if is surjective. *

Let and be continuous g-Bessel sequences for with respect to . In [21], the authors defined an adjointable operator about them as follows:

Theorem 15. *Let be a continuous g-frame for with respect to with bounds and frame operator , and is a continuous g-Bessel sequence for with respect to . Suppose that there exists a number such that for all ,
**
Then is a modular continuous g-Riesz basis for if and only if is a modular continuous g-Riesz basis for . *

*Proof. *For any , we have
So, is bounded below with respect to norm. On the other hand, since
by the above result, is also bounded below with respect to norm, and hence, by Lemma 2, both and are surjective, and furthermore, is invertible. Let and be the synthesis operators of and , respectively. It is easy to check that . Thus, is invertible if and only if is invertible, and consequently, is a modular continuous g-Riesz basis for if and only if is a modular continuous g-Riesz basis for .

#### 3. The Equivalency Relations between Continuous g-Frames in Hilbert -Modules

The definitions of similar and unitary equivalent frames give rise to definitions of similar and unitary equivalent continuous g-frames in Hilbert -modules.

*Definition 16. *Let and be two continuous g-frames for with respect to . One has the following.(1)They are said to be similar or equivalent if there is an adjointable and invertible operator such that for each .(2)They are said to be unitary equivalent if there exists an adjointable and unitary linear operator such that for each .

Theorem 17. *Let and be two continuous g-frames for with respect to with synthesis operators and , respectively. Then the following statements are equivalent:*(1)*there is an adjointable and invertible operator such that for each ; that is, and are similar;*(2)*there exists a constant such that
**for all . Moreover, if (2) holds, then
*

*Proof. *. Suppose that is an adjointable and invertible operator such that for each . If for certain , then we have
Therefore,

On the other hand,
Hence, (32) follows.

. For each , we define an operator as follows:
It is clear that is well defined, and furthermore, is adjointable. A simple calculation shows that its adjoint operator is given by
where is the frame operator of . Since is surjective by Proposition 7, it follows that is also surjective. And (32) implies that is injective, and so is invertible. It remains to establish that for each . For all , we have
That is, . Hence, for each .

For the last statement, the assumptions implies that and for all . If we replace by in the last inequality, we have . Therefore,
This completes the proof.

To complete this section, we generalize the results in [27] for g-frames in Hilbert spaces to continuous g-frames in Hilbert -modules.

Proposition 18. *Let and be two continuous Parseval g-frames for with respect to with synthesis operators and , respectively. Then*(1)* if and only if there exists an adjointable operator ** which preserves inner product such that ** for each **. Conversely, if ** is an adjointable operator which preserves inner product such that ** for each **, then*(2)* if and only if ** and ** are unitary equivalent.*

*Proof. *(1) “⇒”. Assume that . Let us denote and . Since both and are surjective, we know that and . Since and are two continuous Parseval g-frames for , it follows that and are orthogonal projections from onto and , respectively. Let , then, for an arbitrary element of , recalling that , we have
Thus, preserves inner product. Also,
and so,
Note that
it follows that
Hence, for each , and for each as a consequence.

“⇐”. It is obvious.

For the second part of (1), since is an isometry, it follows that

(2) Suppose that , then (41) implies that , and hence, . Thus, is injective, and so, is invertible. Since , it follows that is unitary. For the other implication, let be a unitary linear operator such that for each . Then , and so, .

For the general case, we have the following proposition.

Proposition 19. *Let and be two continuous g-frames for with respect to with synthesis operators and and frame operators and , respectively. Then*(1)* if and only if there exists an adjointable operator ** such that ** for each **;*(2)* if and only if ** and ** are similar.*

*Proof. *(1) “⇒”. Assume that . We already know that and are closed submodules of . Then and , and thus, . It is easy to check that and are both continuous Parseval g-frames. Let us denote by and the synthesis operators of and , respectively. Then and . Therefore, and . By Proposition 18, there exists an adjointable operator such that for each . Hence, the result follows by letting .

“⇐”. It is straightforward.

(2) “⇒”. If , then . By part (2) of Proposition 18, is unitary, and consequently, is invertible.

“⇐”. It is obvious.

#### 4. Stability of Duals of Continuous g-Frames in Hilbert -Modules

The stability of frames is important in practice and is therefore studied widely by many authors. The stability of dual frames is also needed in practice. However, most of the known results on this topic are stated about canonical dual; see [28] for frames in Hilbert spaces and [29, 30] for g-frames in Hilbert spaces. Fortunately, Arefijamaal and Ghasemi [31] presented a stability result for alternate duals of g-frames in Hilbert spaces by observing the difference between an alternate dual and the canonical dual. In what follows, we will generalize their result to alternate duals of continuous g-frames in Hilbert -modules. We start with the following lemma, which shows that the difference between an alternate dual and the canonical dual can be considered as an adjointable operator.

Lemma 20. *Let be a continuous g-frame for with respect to with bounds and the synthesis operator . Then there exists a one-to-one correspondence between the duals of and operator such that . *

*Proof. *Assume first that is a dual continuous g-frame of with bounds and , and let be the frame operator of . Define by
Then is adjointable, that is; . Indeed,
for all . Moreover, we have

Conversely, let and . Take
Since
it follows that is a continuous g-Bessel sequence for with respect to . Furthermore,
Thus, is a dual continuous g-frame of , by Proposition 6.

Theorem 21. *Let and be two continuous g-frames for with respect to with bounds and , respectively. Also, let be a fixed dual of with frame bounds . If is a continuous g-Bessel sequence with Bessel bound , then there exists a dual of such that is also a continuous g-Bessel sequence. *

*Proof. *Let us denote by , and , the synthesis operators and frame operators of and , respectively. By the proof of Lemma 20 we know that there exists with
such that for all . A simple calculation shows that

Let
It is easy to see that is a continuous g-Bessel sequence. Let be the synthesis operator of , then and
Hence, , and furthermore, is invertible. Therefore, every can be represented by
showing that is a dual of . In what follows, we will show that is the desired continuous g-frame.

If we take , then
and so,
Denoting by the synthesis operator of , we have
where is the upper frame bound of . The proof is completed.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11271148). The author thanks the referee(s) and the editor(s) for their valuable comments and suggestions which improved the quality of the paper.