We give a generalization of g-frame in Hilbert C^∗-modules that was introduced by Khosravies then investigated some properties of it by Xiao and Zeng. This generalization is a natural generalization of continuous and discrete g-frames and frame in Hilbert space too. We characterize continuous g-frame g-Riesz in Hilbert C^∗-modules and give some equality and inequality of these frames.

1. Introduction

Frames for Hilbert spaces were first introduced in 1952 by Duffin and Schaeffer [1] for study of nonharmonic Fourier series. They were reintroduced and development in 1986 by Daubechies et al. [2] and popularized from then on. The theory of frames plays an important role in signal processing because of their importance to quantization [3], importance to additive noise [4], as well their numerical stability of reconstruction [4] and greater freedom to capture signal characteristics [5, 6]. See also [7–9]. Frames have been used in sampling theory [10, 11], to oversampled perfect reconstruction filter banks [12], system modelling [13], neural networks [14] and quantum measurements [15]. New applications in image processing [16], robust transmission over the Internet and wireless [17–19], coding and communication [20, 21] were given. For basic results on frames, see [4, 12, 22, 23].

let H be a Hilbert space, and I a set which is finite or countable. A system {f_i} _i∈I⊆H is called a frame for H if there exist the constants A, B > 0 such that

()

for all f ∈ H. The constants A and B are called frame bounds. If A = B we call this frame a tight frame and if A = B = 1 it is called a Parseval frame.

In [24] Sun introduced a generalization of frames and showed that this includes more other cases of generalizations of frame concept and proved that many basic properties can be derived within this more general context.

Let U and V be two Hilbert spaces, and {V_j : j ∈ J} is a sequence of subspaces of V, where J is a subset of Z. L(U, V_j) is the collection of all bounded linear operators from U into V_j. We call a sequence {Λ_j ∈ L(U, V_j) : j ∈ J} a generalized frame, or simply a g-frame, for U with respect to {V_j : j ∈ J} if there are two positive constants A and B such that

()

for all f ∈ U. The constants A and B are called g-frame bounds. If A = B we call this g-frame a tight g-frame, and if A = B = 1 it is called a Parseval g-frame.

On the other hand, the concept of frames especially the g-frames was introduced in Hilbert C^*-modules, and some of their properties were investigated in [25–27]. Frank and Larson [25] defined the standard frames in Hilbert C^*-modules in 1998 and got a series of result for standard frames in finitely or countably generated Hilbert C^*-modules over unital C^*-algebras. As for Hilbert C^*-module, it is a generalization of Hilbert spaces in that it allows the inner product to take values in a C^*-algebra rather than the field of complex numbers. There are many differences between Hilbert C^*-modules and Hilbert spaces. For example, we know that any closed subspace in a Hilbert space has an orthogonal complement, but it is not true for Hilbert C^*-module. And we cannot get the analogue of the Riesz representation theorem of continuous functionals in Hilbert C^*-modules generally. Thus it is more difficult to make a discussion of the theory of Hilbert C^*-modules than that of Hilbert spaces in general. We refer the readers to papers [28, 29] for more details on Hilbert C^*-modules. In [27, 30], the authors made a discussion of some properties of g-frame in Hilbert C^*-module in some aspects.

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 [23] and independently by Ali at al. [31]. These frames are known as continuous frames. Let H be a Hilbert space, and let (M; 𝒮; μ) be a measure space. A continuous frame in H indexed by M is a family h = {h_m ∈ H : m ∈ M} such that

(a)
for any f ∈ H, the function defined by is measurable;
(b)
there is a pair of constants 0 < A, B such that, for any f ∈ H,

()

If M = N and μ is the counting measure, the continuous frame is a frame.

The paper is organized as follows. In Sections 2 and 3 we recall the basic definitions and some notations about continuous g-frames in Hilbert C^*-module; we also give some basic properties of g-frames which we will use in the later sections. In Section 4, we give some characterization for continuous g-frames in Hilbert C^*-modules. In Section 5, we extend some important equalities and inequalities of frame in Hilbert spaces to continuous frames and continuous g-frames in Hilbert C^*-modules.

2. Preliminaries

In the following we review some definitions and basic properties of Hilbert C^*-modules and g-frames in Hilbert C^*-module; we first introduce the definition of Hilbert C^*-modules.

Definition 2.1. Let A be a C^*-algebra with involution ^*. An inner product A-module (or pre-Hilbert A-module) is a complex linear space H which is a left A-module with map 〈·, ·〉:H × H → A which satisfies the following properties:

(1)
〈αf + βg, h〉 = α〈f, h〉 + β〈g, h〉 for all f, g, h ∈ H and α, β ∈ C;
(2)
〈af, g〉 = a〈f, g〉 for all f, g ∈ H and a ∈ A;
(3)
〈f, g〉 = 〈g, f〉 ^* for all f, g ∈ H;
(4)
〈f, f〉≥0 for all f ∈ H and 〈f, f〉 = 0 if and only if f = 0.

For f ∈ H, we define a norm on H by

. Let H is complete with this norm, it is called a Hilbert C^*-module over A or a Hilbert A-module.

An element a of a C^*-algebra A is positive if a^* = a and spectrum of a is a subset of positive real number. We write a ≥ 0 to mean that a is positive. It is easy to see that 〈f, f〉≥0 for every f ∈ H, hence we define |f | = 〈f, f〉 ^1/2.

Frank and Larson [25] defined the standard frames in Hilbert C^*-modules. If H be a Hilbert C^*-module, and I a set which is finite or countable. A system {f_i} _i∈I⊆H is called a frame for H if there exist the constants A, B > 0 such that

()

for all f ∈ H. The constants A and B are called frame bounds.

A. Khosravi and B. Khosravi [27] defined g-frame in Hilbert C^*-module. Let U and V be two Hilbert C^*-module, and {V_i : i ∈ I} is a sequence of subspaces of V, where I is a subset of Z and

is the collection of all adjointable A-linear maps from U into V_i that is, 〈Tf, g〉 = 〈f, T^*g〉 for all f, g ∈ H and

. We call a sequence

a generalized frame, or simply a g-frame, for Hilbert C^*-module U with respect to {V_i : i ∈ I} if there are two positive constants A and B such that

()

for all f ∈ U. The constants A and B are called g-frame bounds. If A = B we call this g-frame a tight g-frame, and if A = B = 1 it is called a Parseval g-frame.

Let (M; 𝒮; μ) be a measure space, let U and V be two Hilbert C^*-modules, {V_m : m ∈ M} is a sequence of subspaces of V, and is the collection of all adjointable A-linear maps from U into V_m.

Definition 2.2. We call a net a continuous generalized frame, or simply a continuous g-frame, for Hilbert C^*-module U with respect to {V_m : m ∈ M} if

(a)
for any f ∈ U, the function defined by is measurable;
(b)
there is a pair of constants 0 < A, B such that, for any f ∈ U,

()

The constants A and B are called continuous g-frame bounds. If A = B we call this continuous g-frame a continuous tight g-frame, and if A = B = 1 it is called a continuous Parseval g-frame. If only the right-hand inequality of (2.3) is satisfied, we call {Λ_m : m ∈ M} the continuous g-Bessel for U with respect to {V_m : m ∈ M} with Bessel bound B.

If M = N and μ is the counting measure, the continuous g-frame for U with respect to {V_m : m ∈ M} is a g-frame for U with respect to {V_m : m ∈ M}.

Let X be a Banach space, (Ω, μ) a measure space, and function f : Ω → X a measurable function. Integral of the Banach-valued function f has defined Bochner and others. Most properties of this integral are similar to those of the integral of real-valued functions for example triangle inequality. The reader is referred to [32, 33] for more details. Because every C^*-algebra and Hilbert C^*-module is a Banach space thus we can use this integral and its properties.

Example 2.3. Let U be a Hilbert C^*-module on C^*-algebra A, and let {f_m : m ∈ M} be a frame for U. Let Λ_m be the functional induced by

()

Then {Λ_m : m ∈ M} is a g-frame for Hilbert C^*-module U with respect to V = V_m = A.

Example 2.4. If ψ ∈ L²(R) is admissible, that is,

()

and, for a, b ∈ R and a ≠ 0

()

then {ψ_a,b} _{a≠0, b∈R} is a continuous frame for L²(R) with respect to R∖{0} × R equipped with the measure da db/a² and, for all f ∈ L²(R),

()

where W_ψ is the continuous wavelet transform defined by

()

For details, see [12, Proposition 11.1.1 and Corollary 11.1.2].

3. Continuous g-Frame Operator and Dual Continuous g-Frame on Hilbert C^*-Algebra

Let {Λ_m : m ∈ M} be a continuous g-frame for U with respect to {V_m : m ∈ M}. Define the continuous g-frame operator S on U by

()

Lemma 3.1 (see [33].)Let (Ω, μ) be a measure space, X and Y are two Banach spaces, λ : X → Y be a bounded linear operator and f : Ω → X measurable function; then

()

Proposition 3.2. The frame operator S is a bounded, positive, selfadjoint, and invertible.

Proof. First we show, S is a selfadjoint operator. By Lemma 3.1 and property (3) of Definition 2.1 for any f, g ∈ U we have

()

It is clear that we have

()

Now we show that S is a bounded operator

()

Inequality (2.3) and equality (3.4) mean that

()

or in the notation from operator theory AI ≤ S ≤ BI; thus S is a positive operator. Furthermore, 0 ≤ A⁻¹S − I ≤ ((B − A)/A)I, and consequently ∥A⁻¹S − I∥≤1; this shows that S is invertible.

Proposition 3.3. Let {Λ_m : m ∈ M} be a continuous g-frame for U with respect to {V_m : m ∈ M} with continuous g-frame operator S with bounds A and B. Then defined by is a continuous g-frame for U with respect to {V_m : m ∈ M} with continuous g-frame operator S⁻¹ with bounds 1/B and 1/A. That is called continuous canonical dual g-frame of {Λ_m : m ∈ M}.

Proof. Let be the continuous g-frame operator associated with that is . Then for f ∈ U,

()

Hence

Since {Λ_m : m ∈ M} is a continuous g-frame for H, then AI ≤ S ≤ BI. On other hand since I and S are selfadjoint and S⁻¹ commutative with I and S, AIS⁻¹ ≤ SS⁻¹ ≤ BIS⁻¹, and hence B⁻¹I ≤ S⁻¹ ≤ A⁻¹I.

Remark 3.4. We have . In other words {Λ_m : m ∈ M} and are dual continuous g-frame with respect to each other.

4. Some Characterizations of Continuous g-Frames in Hilbert C^*-Module

In this section, we will characterize the equivalencies of continuous g-frame in Hilbert C^*-module from several aspects. As for Theorem 4.2, we show that the continuous g-frame is equivalent to which the middle of (2.3) is norm bounded. As for Theorems 4.3 and 4.6, the characterization of g-frame is equivalent to the characterization of bounded operator T.

Lemma 4.1 (see [34].)Let A be a C^*-algebra, U and V two Hilbert A-modules, and . Then the following statements are equivalent:

(1)
T is surjective;
(2)
T^* is bounded below with respect to norm, that is, there is m > 0 such that ∥T^*f∥≥m∥f∥ for all f ∈ U;
(3)
T^* is bounded below with respect to the inner product, that is, there is m^′ > 0 such that 〈T^*f, T^*f〉≥m^′〈f, f〉.

Theorem 4.2. Let for any m ∈ M. Then {Λ_m : m ∈ M} be a continuous g-frame for U with respect to {V_m : m ∈ M} if and only if there exist constants A, B > 0 such that for any f ∈ U

()

Proof. Let {Λ_m : m ∈ M} is a continuous g-frame for U with respect to {V_m : m ∈ M}. Then inequality (4.1) is an immediate result of C^*-algebra theory.

If inequality (4.1) holds, then by Proposition 3.2, 〈S^1/2f, S^1/2f〉 = 〈Sf, f〉 = ∫_M〈Λ_mf, Λ_mf〉dμ(m), hence for any f ∈ U. Now by use of Lemma 4.1, there are constants C, D > 0 such that

()

which implies that {Λ_m : m ∈ M} is a continuous g-frame for U with respect to {V_m : m ∈ M}.

We define

()

For any f = {f_m : m ∈ M} and g = {g_m : m ∈ M}, if the A-valued inner product is defined by 〈f, g〉 = ∫_M〈f_m, g_m〉dμ(m), the norm is defined by ∥f∥ = ∥〈f,f〉∥^1/2, then ⨁_m∈MV_m is a Hilbert A-module (see [28]).

Let be a continuous g-frame for U with respect to {V_m : m ∈ M}, we define synthesis operator T : ⨁_m∈MV_m → U by; for all g = {g_m : m ∈ M} ∈ ⨁_m∈MV_m. So analysis operator is defined for map F : U → ⨁_m∈MV_m by F(f) = {Λ_m : m ∈ M} for any f ∈ U.

Theorem 4.3. A net is a continuous g-frame for U with respect to {V_m : m ∈ M} if and only if synthesis operator T is well defined and surjective.

Proof. Let {Λ_m : m ∈ M} be a continuous g-frame for U with respect to {V_m : m ∈ M}; then operator T is well defined and because

()

For any f ∈ U, by that S is invertible, there exist g ∈ U such that . Since {Λ_m : m ∈ M} is a continuous g-frame for U with respect to {V_m : m ∈ M}, so {λ_mg : m ∈ M} ∈ ⨁_m∈MV_m and , which implies that T is surjective.

Now let T be a well-defined operator. Then for any f ∈ U we have

()

It follow that ∥∫_M〈Λ_mf, Λ_mf〉dμ(m)∥ ≤ ∥f∥²∥T∥².

On the other hand, since T is surjective, by Lemma 4.1, T^* is bounded below, so is invertible. Then for any f ∈ U, we have , so . It is easy to check that

()

Hence

Corollary 4.4. A net is a continuous g-Bessel net for U with respect to {V_m : m ∈ M} if and only if synthesis operator T is well defined and .

Definition 4.5. A continuous g-frame in Hilbert C^*-module U with respect to {V_m : m ∈ M} is said to be a continuous g-Riesz basis if it satisfies that

(1)
λ_m ≠ 0 for any m ∈ M;
(2)
if , then every summand is equal to zero, where {g_m} _m∈K ∈ ⨁_m∈KV_m and K is a measurable subset of M.

Theorem 4.6. A net is a continuous g-Riesz for U with respect to {V_m : m ∈ M} if and only if synthesis operator T is homeomorphism.

Proof. We firstly suppose that is a continuous g-Bessel net for U with respect to {V_m : m ∈ M}. By Theorem 4.3 and that it is g-frame, T is surjective. If for some f = {f_m : m ∈ M} ∈ ⨁_m∈MV_m, according to the definition of continuous g-Riesz basis we have for any m ∈ M, and Λ_m ≠ 0, so f_m = 0 for any m ∈ M, namely f = 0. Hence T is injective.

Now we let the synthesis operator T be homeomorphism. By Theorem 4.3 is a continuous g-frame for U with respect to {V_m : m ∈ M}. It is obviouse that Λ_m ≠ 0 for any m ∈ M. Since T is injective, so if , then f = {f_m : m ∈ M} = 0, so . Therefore is a continuous g-Riesz for U with respect to {V_m : m ∈ M}.

Theorem 4.7. Let be a continuous g-frame for U with respect to {V_m : m ∈ M}, with g-frame bounds A₁, B₁ ≥ 0. Let for any m ∈ M. Then the following are equivalent:

(1)
is a continuous g-frame for U with respect to {V_m : m ∈ M};
(2)
there exists a constant N > 0, such that for any f ∈ U, one has

()

Proof. First we let be a continuous g-frame for U with respect to {V_m : m ∈ M} with bounds A₂, B₂ > 0. Then for any f ∈ U, we have

()

Similarly we can obtain

()

Let N = min {1 + B₂/A₁, 1 + B₁/A₂}; then inequality (4.7) holds.

Next we suppose that inequality (4.7) holds. For any f ∈ U, we have

()

Also we can obtain

()

Next we will introduce a bounded operator L about two g-Bessel sequences in Hilbert C^*-module. The idea is derived from the operator S_VW which was considered for fusion frames by Găvruţa in [35]. In this paper, we will use the operator L to characterize the g-frames of Hilbert C^*-module further. Let

and

be two g-Bessel sequences for U with respect to {V_m : m ∈ M}, with Bessel bounds B₁, B₂ > 0, respectively. Then there exists a well-defined operator

()

As a matter of fact, for any f ∈ U, we have

()

It is easy to know that

and

Theorem 4.8. Let be a continuous g-frame for U with respect to {V_m : m ∈ M}, with g-frame bounds A₁, B₁ ≥ 0. Suppose that is a continuous g-Bessel net for U with respect to {V_m : m ∈ M}. If L is surjective, then {Γ_m : m ∈ M} is a continuous g-frame for U with respect to {V_m : m ∈ M}.

On the contrary, if is a continuous g-Riesz basis for U with respect to {V_m : m ∈ M}, then L is surjective.

Proof. Suppose that is a continuous g-frame for U with respect to {V_m : m ∈ M}. By Theorem 4.3, we can define the synthesis operator T of (4.12). It is easy to check that the adjoint operator of T is analysis operator as follows:

()

for any f ∈ U.

On the other hand, since is a continuous g-Bessel net for U with respect to {V_m : m ∈ M}, by Corollary 4.4 we also can define the corresponding operator .

Hence we have for any f ∈ U, namely, L = QT^*. Since L is surjective, then for any f ∈ U, there exists g ∈ U such that f = Lg = QT^*g, and T^*g ∈ ⨁_m∈MV_m, it follows that Q is surjective. By Theorem 4.3 we know that is a continuous g-frame for U with respect to {V_m : m ∈ M}.

On the contrary, suppose that is a continuous g-Riesz basis and is a continuous g-frame for U with respect to {V_m : m ∈ M}. By Theorem 4.6, T is homeomorphous, so is T^*. By Theorem 4.3 Q is surjective, therefore L = QT^* is surjective.

Theorem 4.9. Let be a continuous g-Riesz basis, is a continuous g-Bessel net for U with respect to {V_m : m ∈ M}. Then {Γ_m : m ∈ M} a continuous g-Riesz basis for U with respect to {V_m : m ∈ M} if and only if L is invertible.

Proof. We first suppose that L is invertible. Since is a g-Riesz basis, is a g-Bessel sequence for U with respect to {V_m : m ∈ M}, by Theorem 4.3 and Corollary 4.4 we can define the operators T, Q mentioned before and T is homeomorphous, hence T^* is also invertible. From the proof of Theorem 4.7 we know that L = QT^*. Since L is invertible, so is Q. By Theorem 4.6 we have that is a g-Riesz basis for U with respect to {V_m : m ∈ M}.

Now we let and be two g-Riesz basis for U with respect to {V_m : m ∈ M}. By Theorem 4.6 both T, Q are invertible, so L = QT^* is invertible too.

5. Some Equalities for Continuous g-frames in Hilbert C^*-Modules

Some equalities for frames involving the real parts of some complex numbers have been established in [36]. These equalities generalized in [30] for g-frames in Hilbert C^*-modules. In this section, we generalize the equalities to a more general form which generalized before equalities and we deduce some equalities for g-frames in Hilbert C^*-modules to alternate dual g-frame.

In [37], the authors verified a longstanding conjecture of the signal processing community: a signal can be reconstructed without information about the phase. While working on efficient algorithms for signal reconstruction, the authors of [38] established the remarkable Parseval frame equality given below.

Theorem 5.1. If {f_j : j ∈ J} is a Parseval frame for Hilbert space H, then for any K ⊂ J and f ∈ H, one has

()

Theorem 5.1 was generalized to alternate dual frames [36]. If {f_j : j ∈ J} is a frame, then frame {g_j : j ∈ J} is called alternate dual frame of {f_j : j ∈ J} if for any f ∈ H, f = ∑_j∈J〈f, g_j〉f_j.

Theorem 5.2. If {f_j : j ∈ J} is a frame for Hilbert space H and {g_j : j ∈ J} is an alternate dual frame of {f_j : j ∈ J}, then for any K ⊂ J and f ∈ H, one has

()

Recently, Zhu and Wu in [39] generalized equality (5.2) to a more general form which does not involve the real parts of the complex numbers.

Theorem 5.3. If {f_j : j ∈ J} is a frame for Hilbert space H and {g_j : j ∈ J} is an alternate dual frame of {f_j : j ∈ J}, then for any K ⊂ J and f ∈ H, one has

()

Now, we extended this equality to continuous g-frames and g-frames in Hilbert C^*-modules and Hilbert spaces. Let H be a Hilbert C^*-module. If is a continuous g-frame for U with respect to {V_m : m ∈ M}, then continuous g-frame is called alternate dual continuous g-frame of if for any f ∈ H, f = ∫_M〈f, Γ_mf〉Λ_mfdμ(m).

Lemma 5.4 (see [30].)Let H be a Hilbert C^*-module. If are two bounded A-linear operators in H and P + Q = I_H, then one has

()

Now, we present main theorem of this section. In following, some result of this theorem for the discrete case will be present.

Theorem 5.5. Let be a continuous g-frame, for Hilbert C^*-module U with respect to {V_m : m ∈ M} and continuous g-frame is alternate dual continuous g-frame of , then for any measurable subset K ⊂ M and f ∈ H, one has

()

Proof. For any measurable subset K ⊂ M, let the operator U_K be defined for any f ∈ H by U_Kf = ∫_K〈f, Γ_mf〉Λ_mfdμ(m).

Then it is easy to prove that the operator U_K is well defined and the integral ∫_K〈f, Γ_mf〉Λ_mfdμ(m) it is finite. By definition alternate dual continuous g-frame . Thus, by Lemma 5.4 we have

()

Hence the theorem holds. The proof is completed.

Corollary 5.6. Let be a discrete g-frame, for Hilbert C^*-module U with respect to {V_j : j ∈ J}, and discrete g-frame is alternate dual discrete g-frame of , then for any subset K ⊂ J and f ∈ H, one has

()

Corollary 5.7. Let {Λ_j ∈ L(U, V_j) : j ∈ J} be a g-frame, for Hilbert space U with respect to {V_j : j ∈ J} and g-frame {Γ_j ∈ L(U, V_j) : j ∈ J} is alternate dual g-frame of {Λ_j ∈ L(U, V_j) : j ∈ J}, then for any measurable subset K ⊂ J and f ∈ H, one has

()

The following results generalize the results in [30] in the case of continuous g-frames.

Lemma 5.8 (see [30].)Let H be a Hilbert C^*-module. If T is a bounded, selfadjoint linear operator and satisfy 〈Tf, f〉 = 0, for all f ∈ H, then T = 0.

Lemma 5.9 (see [30].)Let H be a Hilbert C^*-module. If are two bounded, selfadjoint A-linear operators in H and P + Q = I_H, then one has

()

Theorem 5.10. Let be a continuous g-frame, for Hilbert C^*-module U with respect to {V_m : m ∈ M} and let be the canonical dual continuous g-frame of {Λ_m : m ∈ M}, then for any measurable subset K ⊂ M and f ∈ U, one has

()

Proof. Since S is an invertible, positive operator on U, and , then . Let . By Lemma 5.9, we obtain

()

Replacing f by S^1/2f, then one has

()

On the other hand, we have

()

Associating with (5.12) the proof is finished.

Corollary 5.11. Let {f_m ∈ H : m ∈ M} be a continuous frame for Hilbert C^*-module H with canonical dual frame , then for any measurable subset K ⊂ M and f ∈ H, one has

()

Proof. For any f ∈ U, if we let Λ_mf = 〈f, f_j〉 in Theorem 5.10, then we get the conclusion.

Theorem 5.12. Let be a continuous Parseval g-frame, for Hilbert C^*-module U with respect to {V_m : m ∈ M}, then for any measurable subset K ⊂ M and f ∈ U, one has

()

Proof. Since is a continuous Parseval g-frame in Hilbert C^*-module U with respect to {V_m : m ∈ M}, then for any f ∈ U, we have

()

Hence for any f ∈ U, we have 〈(S − I_U)f, f〉 = 0. Let T = S − I_U. Since S is bounded, selfadjoint, then

, so T is also bounded, selfadjoint. By Lemma 5.8, we have T = 0, namely, S = I_U, so

. From (5.16), then we have that: for any measurable subset K ⊂ M and f ∈ U,

()

Combining (5.16) and Theorem 5.10, we get the result.

Corollary 5.13. Let {f_m ∈ H : m ∈ M} be a continuous Parseval frame for Hilbert C^*-module H, then for any measurable subset K ⊂ M and f ∈ H, one has

()

Corollary 5.14. Let be a continuous λ-tight g-frame, for Hilbert C^*-module U with respect to {V_m : m ∈ M}, then for any measurable subset K ⊂ M and f ∈ U, one has

()

Proof. Since {Λ_m : m ∈ M} is a continuous λ-tight g-frame, then {Λ_m : m ∈ M} is a continuous g-Parseval frame, by Theorem 5.12 we know that the conclusion holds.

Corollary 5.15. Let {f_m ∈ H : m ∈ M} be a continuous λ-tight frame for Hilbert C^*-module H then for any measurable subset K ⊂ M and f ∈ H, one has

()

Corollary 5.16. Let be a continuous λ-tight g-frame, for Hilbert C^*-module U with respect to {V_m : m ∈ M}, then for any measurable subset K, L ⊂ M, K ⋂ L = ϕ and f ∈ U, one has

()

Proof. Since for any f ∈ U, by Corollary 5.14, we get

()

Acknowledgment

The authors would like to give a special thanks to the referee(s) and the editor(s) for their valuable comments and suggestions which improved the presentation of the paper.

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Continuous g-Frame in Hilbert C^∗-Modules

Abstract

1. Introduction

2. Preliminaries

3. Continuous g-Frame Operator and Dual Continuous g-Frame on Hilbert C^*-Algebra

4. Some Characterizations of Continuous g-Frames in Hilbert C^*-Module

5. Some Equalities for Continuous g-frames in Hilbert C^*-Modules

Acknowledgment

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Continuous g-Frame in Hilbert C∗-Modules

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Comment on “Continuous g-Frame in Hilbert C*-Modules”

Abstract

1. Introduction

2. Preliminaries

3. Continuous g-Frame Operator and Dual Continuous g-Frame on Hilbert C*-Algebra

4. Some Characterizations of Continuous g-Frames in Hilbert C*-Module

5. Some Equalities for Continuous g-frames in Hilbert C*-Modules

Acknowledgment

References

Citing Literature

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Continuous g-Frame in Hilbert C^∗-Modules

Comment on “Continuous g-Frame in Hilbert C^*-Modules”

3. Continuous g-Frame Operator and Dual Continuous g-Frame on Hilbert C^*-Algebra

4. Some Characterizations of Continuous g-Frames in Hilbert C^*-Module

5. Some Equalities for Continuous g-frames in Hilbert C^*-Modules