Continuous g-Frame in Hilbert 𝐶^∗-Modules

Abstract

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 𝐴𝑓²≤_𝑖∈𝐼𝑓,𝑓_𝑖²≤𝐵𝑓² (1.1) 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 that𝐴𝑓²≤_𝑖∈𝐽Λ_𝑗𝑓²≤𝐵𝑓² (1.2) 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 𝑓∶𝑀→𝐶 defined by 𝑓(𝑚)=⟨ℎ_𝑚,𝑓⟩ is measurable;(b)there is a pair of constants 0<𝐴,𝐵 such that, for any 𝑓∈𝐻,𝐴𝑓²_𝐻≤𝑓²_𝐿2(𝜇)≤𝐵𝑓²_𝐻. (1.3) 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∈𝐻and𝛼,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 ‖𝑓‖_𝐻=‖⟨𝑓,𝑓⟩‖_𝐴^1/2. 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 𝐴𝑓,𝑓≤_𝑖∈𝐼𝑓,𝑓_𝑖𝑓_𝑖,𝑓≤𝐵𝑓,𝑓 (2.1) 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 End^∗_𝐴(𝑈,𝑉_𝑖) 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 𝑇∈End^∗_𝐴(𝑈,𝑉_𝑖). We call a sequence {Λ_𝑖∈End^∗_𝐴(𝑈,𝑉_𝑖)∶𝑖∈𝐼} 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 that𝐴𝑓,𝑓≤_𝑖∈𝐼Λ_𝑖𝑓,Λ_𝑖𝑓≤𝐵𝑓,𝑓 (2.2) 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 End^∗_𝐴(𝑈,𝑉_𝑚) is the collection of all adjointable 𝐴-linear maps from 𝑈 into 𝑉_𝑚.

Definition 2.2. We call a net {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 𝑓∶𝑀→𝑉_𝑚 defined by 𝑓(𝑚)=Λ_𝑚𝑓 is measurable;(b)there is a pair of constants 0<𝐴,𝐵 such that, for any 𝑓∈𝑈,𝐴𝑓,𝑓≤_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)≤𝐵𝑓,𝑓. (2.3) 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 Λ_𝑚𝑓=𝑓,𝑓_𝑚,∀𝑓∈𝑈. (2.4) 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, 𝐶_𝜓∶=^∞_−∞𝜓𝛾²𝛾𝑑𝛾<∞. (2.5) and, for 𝑎,𝑏∈𝐑 and 𝑎≠0𝜓_𝑎,𝑏(𝑥)=𝑇_𝑏𝐷_𝑎𝜓(𝑥)=1|𝑎|𝜓𝑥−𝑏𝑎,∀𝑥∈𝐑, (2.6) 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 𝑑𝑎𝑑𝑏/𝑎² and, for all 𝑓∈𝐿²(𝐑), 𝑓(𝑥)=1𝐶_𝜓^∞_−∞𝑊_𝜓𝑓(𝑎,𝑏)𝜓_𝑎,𝑏(𝑥)𝑑𝑎𝑑𝑏𝑎², (2.7) where 𝑊_𝜓 is the continuous wavelet transform defined by 𝑊_𝜓𝑓(𝑎,𝑏)=𝑓,𝜓_𝑎,𝑏=^∞_−∞𝑓(𝑥)1|𝑎|𝜓𝑥−𝑏𝑎𝑑𝑥. (2.8) 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 𝑆𝑓=_𝑀Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚). (3.1)

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 𝜆_Ω𝑓𝑑𝜇=_Ω𝜆𝑓𝑑𝜇. (3.2)

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 𝑆𝑓,𝑔=_𝑀Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚),𝑔=_𝑀Λ^∗_𝑚Λ_𝑚(𝑚)=𝑓,𝑔𝑑𝜇_𝑀𝑓,Λ^∗_𝑚Λ_𝑚𝑔𝑑𝜇(𝑚)=𝑓,_𝑀Λ^∗_𝑚Λ_𝑚(𝑚=.𝑔𝑑𝜇)𝑓,𝑆𝑔 (3.3) It is clear that we have 𝑆𝑓,𝑓=_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚). (3.4) Now we show that 𝑆 is a bounded operator ‖𝑆‖=sup_𝑓≤1𝑆𝑓,𝑓=sup_𝑓≤1𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)≤𝐵. (3.5) Inequality (2.3) and equality (3.4) mean that 𝐴𝑓,𝑓≤𝑆𝑓,𝑓≤𝐵𝑓,𝑓 (3.6) 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 {Λ_𝑚∶𝑚∈𝑀} defined by Λ_𝑚=Λ_𝑚𝑆⁻¹ 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 𝑆 be the continuous g-frame operator associated with {Λ_𝑚∶𝑚∈𝑀} that is 𝑆𝑓=∫_𝑀Λ_𝑚^∗Λ_𝑚𝑓𝑑𝜇(𝑚). Then for 𝑓∈𝑈, 𝑆𝑆𝑓=_𝑀𝑆Λ_𝑚^∗Λ_𝑚𝑓𝑑𝜇(𝑚)=_𝑀𝑆𝑆⁻¹Λ^∗_𝑚Λ_𝑚𝑆⁻¹(𝑚)=𝑓𝑑𝜇_𝑀Λ^∗_𝑚Λ_𝑚𝑆⁻¹𝑓𝑑𝜇(𝑚)=𝑆𝑆⁻¹𝑓=𝑓. (3.7) Hence 𝑆=𝑆⁻¹.
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 Λ_𝑚𝑆⁻¹=Λ_𝑚𝑆⁻¹𝑆=Λ_𝑚. In other words {Λ_𝑚∶𝑚∈𝑀} and {Λ_𝑚∶𝑚∈𝑀} 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 𝑇∈End^∗_𝐴(𝑈,𝑉). 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 Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚) 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∈𝑈𝐴𝑓²≤_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)≤𝐵𝑓². (4.1)

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 √𝐴‖𝑓‖≤‖𝑆^1/2𝑓‖≤√𝐵‖𝑓‖ for any 𝑓∈𝑈. Now by use of Lemma 4.1, there are constants 𝐶,𝐷>0 such that 𝐶𝑓,𝑓≤_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)≤𝐷𝑓,𝑓, (4.2) which implies that {Λ_𝑚∶𝑚∈𝑀} is a continuous g-frame for 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀}.

We define_𝑚∈𝑀𝑉_𝑚=𝑔=𝑔_𝑚∶𝑔_𝑚∈𝑉_𝑚,_𝑀𝑔_𝑚²𝑑𝜇(𝑚)<∞. (4.3)

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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} be a continuous g-frame for 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀}, we define synthesis operator 𝑇∶⨁_𝑚∈𝑀𝑉_𝑚→𝑈 by; 𝑇(𝑔)=∫_𝑀Λ^∗_𝑚𝑔_𝑚𝑑𝜇(𝑚) 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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 ‖𝑇‖≤√𝐵 because 𝑇𝑔²=_𝑀Λ^∗_𝑚𝑔_𝑚𝑑𝜇(𝑚)²=sup_{𝑓∈𝑈,‖𝑓‖=1𝑀}Λ^∗_𝑚𝑔_𝑚𝑑𝜇(𝑚),𝑓=sup_{𝑓𝑓∈𝑈,=1𝑀}𝑔_𝑚,Λ_𝑚𝑓𝑑𝜇(𝑚)≤sup_{𝑓𝑓∈𝑈,=1𝑀}𝑔_𝑚,𝑔_𝑚𝑑𝜇(𝑚)_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)≤𝐵_𝑀𝑔_𝑚,𝑔_𝑚𝑑𝜇(𝑚). (4.4)
For any 𝑓∈𝑈, by that 𝑆 is invertible, there exist 𝑔∈𝑈 such that 𝑓=𝑆𝑔=∫_𝑀𝜆^∗_𝑚𝜆_𝑚𝑔𝑑𝜇(𝑚). 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 𝑇({𝜆_𝑚𝑔}_𝑚∈𝑀)=∫_𝑀𝜆^∗_𝑚𝜆_𝑚𝑔𝑑𝜇(𝑚)=𝑓, which implies that 𝑇 is surjective.
Now let 𝑇 be a well-defined operator. Then for any 𝑓∈𝑈 we have _𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)=_𝑀𝑓,Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚)=𝑓,_𝑀Λ^∗_𝑚Λ_𝑚(𝑚≤𝑓𝑓𝑑𝜇)_𝑀Λ^∗_𝑚Λ_𝑚(𝑚=𝑓𝑇Λ𝑓𝑑𝜇)_𝑚𝑓_𝑚∈𝑀≤𝑓‖𝑇‖{Λ_𝑚𝑓}_𝑚∈𝑀≤𝑓‖𝑇‖_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)^1/2. (4.5) 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 𝑇^∗|_𝑅(𝑇^∗₎ is invertible. Then for any 𝑓∈𝑈, we have (𝑇^∗|_𝑅(𝑇^∗₎)⁻¹𝑇^∗𝑓=𝑓, so ‖𝑓‖²≤‖𝑇^∗𝑓‖²‖(𝑇^∗|_𝑅(𝑇^∗₎)⁻¹‖². It is easy to check that 𝑇^∗∶𝑈⟶_𝑚∈𝑀𝑉_𝑚,𝑇^∗𝑓=Λ_𝑚𝑇∶𝑚∈𝑀_∗𝑓²={Λ_𝑚}_𝑚∈𝑀²=_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚). (4.6) Hence ‖𝑓‖²≤‖∫_𝑀⟨Λ_𝑚𝑓,Λ_𝑚𝑓⟩𝑑𝜇(𝑚)‖‖(𝑇^∗|_𝑅(𝑇^∗₎)⁻¹‖².

Corollary 4.4. A net {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 ‖𝑇‖≤√𝐷.

Definition 4.5. A continuous g-frame {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 ∫_𝐾Λ^∗_𝑚𝑔_𝑚𝑑𝜇(𝑚)=0, then every summand Λ^∗_𝑚𝑔_𝑚 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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 𝑇𝑓=∫_𝑀Λ^∗_𝑚𝑓_𝑚𝑑𝜇(𝑚)=0 for some 𝑓={𝑓_𝑚∶𝑚∈𝑀}∈⨁_𝑚∈𝑀𝑉_𝑚, according to the definition of continuous g-Riesz basis we have Λ^∗_𝑚𝑓_𝑚=0 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  {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 𝑇𝑓=∫_𝑀Λ^∗_𝑚𝑓_𝑚𝑑𝜇(𝑚)=0, then 𝑓={𝑓_𝑚∶𝑚∈𝑀}=0, so Λ^∗_𝑚𝑓_𝑚=0. Therefore {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} is a continuous g-Riesz for 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀}.

Theorem 4.7. Let {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} be a continuous g-frame for 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀}, with g-frame bounds 𝐴₁,𝐵₁≥0. Let Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚) for any 𝑚∈𝑀. Then the following are equivalent:(1){Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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_𝑀Λ_𝑚−Γ_𝑚𝑓,Λ_𝑚−Γ_𝑚𝑓𝑑𝜇(𝑚)≤𝑁min_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚),_𝑀Γ_𝑚𝑓,Γ_𝑚𝑓𝑑𝜇(𝑚). (4.7)

Proof. First we let {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 _𝑀Λ_𝑚−Γ_𝑚𝑓,Λ_𝑚−Γ_𝑚𝑓(𝑚=Λ𝑑𝜇)_𝑚−Γ_𝑚𝑓_𝑚∈𝑀²≤Λ_𝑚𝑓_𝑚∈𝑀²+Γ_𝑚𝑓_𝑚∈𝑀²=_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+_𝑀Γ_𝑚𝑓,Γ_𝑚𝑓(𝑚≤𝑑𝜇)_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+𝐵₂𝑓²≤_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓(𝑚+𝐵𝑑𝜇)²𝐴_1𝑀Λ_𝑚𝑓,Λ_𝑚𝑓(𝑚=𝐵𝑑𝜇)1+²𝐴_1𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚). (4.8) Similarly we can obtain _𝑀Λ_𝑚−Γ_𝑚𝑓,Λ_𝑚−Γ_𝑚𝑓(𝑚≤𝐵𝑑𝜇)1+¹𝐴_2𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚). (4.9) 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 𝐴₁𝑓²≤_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)={Λ_𝑚𝑓}_𝑚∈𝑀²≤Λ_𝑚−Γ_𝑚𝑓_𝑚∈𝑀²+Γ_𝑚𝑓_𝑚∈𝑀²=_𝑀Λ_𝑚−Γ_𝑚𝑓,Λ_𝑚−Γ_𝑚𝑓𝑑𝜇(𝑚)+_𝑀Γ_𝑚𝑓,Γ_𝑚𝑓𝑑𝜇(𝑚)≤𝑁_𝑀Γ_𝑚𝑓,Γ_𝑚𝑓𝑑𝜇(𝑚)+_𝑀Γ_𝑚𝑓,Γ_𝑚𝑓(𝑚=(𝑑𝜇)𝑁+1)_𝑀Γ_𝑚𝑓,Γ_𝑚𝑓𝑑𝜇(𝑚). (4.10)
Also we can obtain _𝑀Γ_𝑚𝑓,Γ_𝑚𝑓𝑑𝜇(𝑚)≤Λ_𝑚𝑓_𝑚∈𝑀²+Λ_𝑚−Γ_𝑚𝑓_𝑚∈𝑀²=_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+_𝑀Λ_𝑚−Γ_𝑚𝑓,Λ_𝑚−Γ_𝑚𝑓𝑑𝜇(𝑚)≤𝑁_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓(𝑚=(𝑑𝜇)𝑁+1)_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)≤𝐵₁(𝑁+1)𝑓². (4.11)

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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} and {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 operator𝐿∶𝑈⟶𝑈,𝐿𝑓=_𝑀Γ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚)∀𝑓∈𝑈. (4.12) As a matter of fact, for any 𝑓∈𝑈, we have _𝑀Γ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚)²=sup_{𝑔𝑔∈𝑈,=1𝑀}Γ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚),𝑔²=sup_{𝑔𝑔∈𝑈,=1𝑀}Λ_𝑚𝑓,Γ_𝑚𝑔𝑑𝜇(𝑚)²≤sup_{𝑔𝑔∈𝑈,=1𝑀}Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)_𝑀Γ_𝑚𝑔,Γ_𝑚𝑔𝑑𝜇(𝑚)≤𝐵_2𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚). (4.13) It is easy to know that 𝐿^∗𝑓=∫_𝑀Λ^∗_𝑚Γ_𝑚𝑓𝑑𝜇(𝑚) and ‖𝐿‖≤𝐵₁𝐵₂.

Theorem 4.8. Let {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} be a continuous g-frame for 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀}, with g-frame bounds 𝐴₁,𝐵₁≥0. Suppose that {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} is a continuous g-Riesz basis for 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀}, then 𝐿 is surjective.

Proof. Suppose that {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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: 𝑇^∗∶𝑈⟶_𝑚∈𝑀𝑉_𝑚by𝑇^∗𝑓=Λ_𝑚∶𝑚∈𝑀 (4.14) for any 𝑓∈𝑈.
On the other hand, since {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 𝑄∶⨁_𝑚∈𝑀𝑉_𝑚→𝑈,𝑄(𝑔)=∫_𝑀Γ^∗_𝑚(𝑔_𝑚)𝑑𝜇(𝑚).
Hence we have 𝐿𝑓=∫_𝑀Γ^∗_𝑚Λ𝑓𝑑𝜇(𝑚)=𝑄𝑇^∗𝑓 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 {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} is a continuous g-frame for 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀}.
On the contrary, suppose that {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} is a continuous g-Riesz basis and {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} be a continuous g-Riesz basis, {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} is a g-Riesz basis, {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} is a g-Riesz basis for 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀}.
Now we let {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} and {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 _𝑗∈𝐾𝑓,𝑓_𝑗²−_𝑗∈𝐾𝑓,𝑓_𝑗𝑓_𝑗²=_{𝑗∈𝐾𝑐}𝑓,𝑓_𝑗²−_{𝑗∈𝐾𝑐}𝑓,𝑓_𝑗𝑓_𝑗². (5.1)

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 Re_𝑗∈𝐾𝑓,𝑔_𝑗𝑓,𝑓_𝑗−_𝑗∈𝐾𝑓,𝑔_𝑗𝑓_𝑗²=Re_{𝑗∈𝐾𝑐}𝑓,𝑔_𝑗𝑓,𝑓_𝑗−_{𝑗∈𝐾𝑐}𝑓,𝑔_𝑗𝑓_𝑗². (5.2)

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 _𝑗∈𝐾𝑓,𝑔_𝑗𝑓,𝑓_𝑗−_𝑗∈𝐾𝑓,𝑔_𝑗𝑓_𝑗²=_{𝑗∈𝐾𝑐}𝑓,𝑔_𝑗𝑓,𝑓_𝑗−_{𝑗∈𝐾𝑐}𝑓,𝑔_𝑗𝑓_𝑗². (5.3)

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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} is a continuous g-frame for 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀}, then continuous g-frame {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} is called alternate dual continuous g-frame of {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 𝑃,𝑄∈End^∗_𝐴(𝐻,𝐻) are two bounded 𝐴-linear operators in 𝐻 and 𝑃+𝑄=𝐼_𝐻, then one has 𝑃−𝑃^∗𝑃=𝑄^∗−𝑄^∗𝑄. (5.4)

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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} be a continuous g-frame, for Hilbert 𝐶^∗-module 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀} and continuous g-frame {Γ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} is alternate dual continuous g-frame of {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀}, then for any measurable subset 𝐾⊂𝑀 and 𝑓∈𝐻, one has _𝐾⟨𝑓,Γ_𝑚𝑓⟩⟨𝑓,Λ_𝑚𝑓⟩^∗𝑑𝜇(𝑚)−_𝐾𝑓,Γ_𝑚𝑓Λ_𝑚𝑓𝑑𝜇(𝑚)²=_𝐾𝑐𝑓,Γ_𝑚𝑓𝑓,Λ_𝑚𝑓^∗𝑑𝜇(𝑚)^∗−_{𝑗∈𝐾𝑐}𝑓,Γ_𝑚𝑓Λ_𝑚𝑓𝑑𝜇(𝑚)². (5.5)

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 𝑈_𝐾+𝑈_𝐾^𝑐=1. Thus, by Lemma 5.4 we have _𝐾𝑓,Γ_𝑚𝑓𝑓,Λ_𝑚𝑓^∗𝑑𝜇(𝑚)−_𝐾𝑓,Γ_𝑚𝑓Λ_𝑚𝑓𝑑𝜇(𝑚)²=_𝐾𝑓,Γ_𝑚𝑓Λ_𝑚𝑓,𝑓𝑑𝜇(𝑚)−𝑈_𝐾𝑓,𝑈_𝐾𝑓=𝑈_𝐾𝑓,𝑓−𝑈^∗_𝐾𝑈_𝐾𝑓,𝑓=𝑈^∗_𝐾^𝑐𝑓,𝑓−𝑈^∗_𝐾^𝑐𝑈_𝐾^𝑐=𝑓,𝑓𝑓,𝑈_𝐾^𝑐𝑓−𝑈_𝐾^𝑐𝑓,𝑈_𝐾^𝑐𝑓=_𝐾𝑐𝑓,Γ_𝑚𝑓𝑓,Λ_𝑚𝑓^∗𝑑𝜇(𝑚)^∗−_{𝑗∈𝐾𝑐}𝑓,Γ_𝑚𝑓Λ_𝑚𝑓𝑑𝜇(𝑚)². (5.6) Hence the theorem holds. The proof is completed.

Corollary 5.6. Let {Λ_𝑗∈End^∗_𝐴(𝑈,𝑉_𝑗)∶𝑗∈𝐽} be a discrete g-frame, for Hilbert 𝐶^∗-module 𝑈 with respect to {𝑉_𝑗∶𝑗∈𝐽}, and discrete g-frame {Γ_𝑗∈End^∗_𝐴(𝑈,𝑉_𝑗)∶𝑗∈𝐽} is alternate dual discrete g-frame of {Λ_𝑗∈End^∗_𝐴(𝑈,𝑉_𝑗)∶𝑗∈𝐽}, then for any subset 𝐾⊂𝐽 and 𝑓∈𝐻, one has _𝑗∈𝐾𝑓,Γ_𝑗𝑓𝑓,Λ_𝑗𝑓^∗−_𝑗∈𝐾𝑓,Γ_𝑗𝑓Λ_𝑗𝑓²=_{𝑗∈𝐾𝑐}𝑓,Γ_𝑗𝑓𝑓,Λ_𝑗𝑓^∗∗−_{𝑗∈𝐾𝑐}𝑓,Γ_𝑗𝑓Λ_𝑗𝑓². (5.7)

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 _𝑗∈K𝑓,Γ_𝑗𝑓𝑓,Λ_𝑗𝑓−_𝑗∈𝐾𝑓,Γ_𝑗𝑓Λ_𝑗𝑓²=_{𝑗∈𝐾𝑐}𝑓,Γ_𝑗𝑓𝑓,Λ_𝑗𝑓−_{𝑗∈𝐾𝑐}𝑓,Γ_𝑗𝑓Λ_𝑗𝑓². (5.8)

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 𝑃,𝑄∈End^∗_𝐴(𝐻,𝐻) are two bounded, selfadjoint 𝐴-linear operators in 𝐻 and 𝑃+𝑄=𝐼_𝐻, then one has 𝑃𝑓,𝑓+𝑄𝑓²=𝑄𝑓,𝑓+𝑃𝑓²≥34𝑓,𝑓. (5.9)

Theorem 5.10. Let {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} be a continuous g-frame, for Hilbert 𝐶^∗-module 𝑈 with respect to {𝑉_𝑚∶𝑚∈𝑀} and let {Λ_𝑚∶𝑚∈𝑀} be the canonical dual continuous g-frame of {Λ_𝑚∶𝑚∈𝑀}, then for any measurable subset 𝐾⊂𝑀 and 𝑓∈𝑈, one has _𝐾Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+_𝑀Λ_𝑚𝑆_𝐾^𝑐𝑓,Λ_𝑚𝑆_𝐾^𝑐𝑓(𝑚)=𝑑𝜇_𝐾𝑐Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+_𝑀Λ_𝑚𝑆_𝐾𝑓,Λ_𝑚𝑆_𝐾𝑓(𝑚)≥3𝑑𝜇4_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚). (5.10)

Proof. Since 𝑆 is an invertible, positive operator on 𝑈, and 𝑆_𝐾+𝑆_K^𝑐=𝑆, then 𝑆^−1/2𝑆_𝐾𝑆^−1/2+𝑆^−1/2𝑆_𝐾^𝑐𝑆^−1/2=𝐼_𝑈. Let 𝑃=𝑆^−1/2𝑆_𝐾𝑆^−1/2,𝑄=𝑆^−1/2𝑆_𝐾^𝑐𝑆^−1/2. By Lemma 5.9, we obtain 𝑆^−1/2𝑆_𝐾𝑆^−1/2𝑓,𝑓+𝑆^−1/2𝑆_𝐾^𝑐𝑆^−1/2𝑓²=𝑆^−1/2𝑆_𝐾^𝑐𝑆^−1/2𝑓,𝑓+𝑆^−1/2𝑆_𝐾𝑆^−1/2𝑓²≥34𝑓,𝑓. (5.11) Replacing 𝑓 by 𝑆^1/2𝑓, then one has 𝑆_𝐾𝑓,𝑓+𝑆⁻¹𝑆_𝐾^𝑐𝑓,𝑆_𝐾^𝑐𝑓=𝑆_𝐾^𝑐𝑓,𝑓+𝑆⁻¹𝑆_𝐾𝑓,𝑆_𝐾𝑓≥34𝑆𝑓,𝑓. (5.12) On the other hand, we have 𝑆_𝐾𝑓,𝑓=_𝐾Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚),𝑓=_𝐾Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇_𝐾(𝑚),Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)=_𝐾Λ_𝑚𝑆⁻¹𝑓,Λ_𝑚𝑆⁻¹𝑓𝑑𝜇(𝑚)=_𝐾Λ^∗_𝑚Λ_𝑚𝑆⁻¹𝑓,Λ_𝑚𝑆⁻¹𝑓(𝑚)=𝑑𝜇_𝐾Λ^∗_𝑚Λ_𝑚𝑆⁻¹𝑓𝑑𝜇(𝑚),Λ_𝑚𝑆⁻¹𝑓=𝑆𝑆⁻¹𝑓,𝑆⁻¹𝑓=𝑓,𝑆⁻¹𝑓=𝑆⁻¹𝑓,𝑓. (5.13)
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 {𝑓_𝑚∈𝐻∶𝑚∈𝑀}, then for any measurable subset 𝐾⊂𝑀 and 𝑓∈𝐻, one has _𝐾𝑓,𝑓_𝑚𝑓_𝑚,𝑓𝑑𝜇(𝑚)+_𝑀𝑆_𝐾^𝑐𝑓,𝑓_𝑗𝑓_𝑗,𝑆_𝐾^𝑐𝑓(𝑚)=𝑑𝜇_𝐾𝑐𝑓,𝑓_𝑚𝑓_𝑚,𝑓𝑑𝜇(𝑚)+_𝑀𝑆_𝐾𝑓,𝑓_𝑗𝑓_𝑗,𝑆_𝐾𝑓𝑑𝜇(𝑚). (5.14)

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 {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 _𝐾Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+_𝐾𝑐Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚)²=_𝐾𝑐Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+_𝐾Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚)²≥34𝑓,𝑓. (5.15)

Proof. Since {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 _𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)=𝑓,𝑓. (5.16) So 𝑆𝑓,𝑓=_𝑀Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚),𝑓=_𝑀Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)=𝑓,𝑓. (5.17) Hence for any 𝑓∈𝑈, we have ⟨(𝑆−𝐼_𝑈)𝑓,𝑓⟩=0. Let 𝑇=𝑆−𝐼_𝑈. Since 𝑆 is bounded, selfadjoint, then 𝑇^∗=(𝑆−𝐼_𝑈)^∗=𝑆^∗−𝐼^∗_𝑈=𝑆−𝐼_𝑈=𝑇, so 𝑇 is also bounded, selfadjoint. By Lemma 5.8, we have 𝑇=0, namely, 𝑆=𝐼_𝑈, so Λ_𝑚=Λ_𝑚𝑆⁻¹=Λ_𝑚. From (5.16), then we have that: for any measurable subset 𝐾⊂𝑀 and 𝑓∈𝑈, _𝑀Λ_𝑚𝑆_𝐾𝑓,Λ_𝑚𝑆_𝐾𝑓𝑑𝜇(𝑚)=_𝑀Λ_𝑚𝑆_𝐾𝑓,Λ_𝑚𝑆_𝐾𝑓(𝑚)=𝑆𝑑𝜇_𝐾𝑓,𝑆_𝐾𝑓=_𝐾Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚)²,_𝑀Λ_𝑚𝑆_𝐾^𝑐𝑓,Λ_𝑚𝑆_𝐾^𝑐𝑓𝑑𝜇(𝑚)=_𝑀Λ_𝑚𝑆_𝐾^𝑐𝑓,Λ_𝑚𝑆_𝐾^𝑐𝑓(𝑚)=𝑆𝑑𝜇_𝐾𝑐𝑓,𝑆_𝐾^𝑐𝑓=_𝐾𝑐Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚)². (5.18) 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 _𝐾𝑓,𝑓_𝑚𝑓_𝑚,𝑓𝑑𝜇(𝑚)+_𝐾𝑐𝑓,𝑓_𝑗𝑓_𝑗𝑑𝜇(𝑚)²=_𝐾𝑐𝑓,𝑓_𝑚𝑓_𝑚,𝑓𝑑𝜇(𝑚)+_𝐾𝑓,𝑓_𝑗𝑓_𝑗𝑑𝜇(𝑚)². (5.19)

Corollary 5.14. Let {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 𝜆_𝐾Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+_𝐾𝑐Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚)²=𝜆_𝐾𝑐Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+_𝐾Λ^∗_𝑚Λ_𝑚𝑓𝑑𝜇(𝑚)². (5.20)

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 𝜆_𝐾𝑓,𝑓_𝑚𝑓_𝑚,𝑓𝑑𝜇(𝑚)+_𝐾𝑐𝑓,𝑓_𝑗𝑓_𝑗𝑑𝜇(𝑚)²=𝜆_𝐾𝑐𝑓,𝑓_𝑚𝑓_𝑚,𝑓𝑑𝜇(𝑚)+_𝐾𝑓,𝑓_𝑗𝑓_𝑗𝑑𝜇(𝑚)². (5.21)

Corollary 5.16. Let {Λ_𝑚∈End^∗_𝐴(𝑈,𝑉_𝑚)∶𝑚∈𝑀} 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 𝑆_𝐾⋃𝐿𝑓²−𝑆_𝐾^𝑐_⧵𝐿𝑓²=𝑆_𝐾𝑓²−𝑆_𝐾^𝑐𝑓²+2𝜆_𝐿Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚). (5.22)

Proof. Since for any 𝑓∈𝑈, by Corollary 5.14, we get 𝑆_𝐾⋃𝐿𝑓|²−𝑆_𝐾^𝑐_⧵𝐿𝑓|²=𝜆_𝐾⋃𝐿Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)−𝜆_{𝐾𝑐⧵𝐿}Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)=𝜆_𝐾Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+𝜆_𝐿Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)−𝜆_𝐾𝑐Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚)+𝜆_𝐿Λ_𝑚𝑓,Λ_𝑚𝑓(𝑚)=𝑆𝑑𝜇_𝐾𝑓²−𝑆_𝐾^𝑐𝑓²+2𝜆_𝐿Λ_𝑚𝑓,Λ_𝑚𝑓𝑑𝜇(𝑚). (5.23)

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.