# Equivalency Relations between Continuous g-Frames and Stability of Alternate Duals of Continuous g-Frames in HilbertC*-Modules

Authors: Zhong-Qi Xiang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We introduce the modular continuous g-Riesz basis to improve one existing result for continuous g-Riesz basis in HilbertC*-modules, and then we study the equivalency relations between continuous g-frames in Hilbert C*-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 C*-modules.


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## Body

## 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 HilbertC*-modules, and some of their properties were investigated in [11–13]. As for Hilbert C*-module, it is a generalization of Hilbert spaces by allowing the inner product to take values in a C*-algebra rather than the field of complex numbers. Note that the theory of Hilbert C*-modules is quite different from that of Hilbert spaces. Unlike Hilbert space cases, not every closed submodule of a Hilbert C*-module is complemented. Moreover, the well-known Riesz representation theorem for continuous functionals in Hilbert spaces does not hold in Hilbert C*-modules, which implies that not all bounded linear operators on Hilbert C*-modules are adjointable. It should also be remarked that, due to the complexity of the C*-algebras involved in the Hilbert C*-modules and the fact that some useful techniques available in Hilbert spaces are either absent or unknown in Hilbert C*-modules, the problems about frames and g-frames for Hilbert C*-modules are more complicated than those for Hilbert spaces. This makes the study of the frames for Hilbert C*-modules important and interesting. The properties of g-frames for Hilbert C*-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 HilbertC*-modules, which were proposed by Kouchi and Nazari in [21], are an extension to g-frames in Hilbert C*-modules and continuous frames in Hilbert spaces, and they made a discussion of some properties of continuous g-frames in Hilbert C*-modules in some aspects. The purpose of this paper is to further investigate the properties of continuous g-frames in Hilbert C*-modules.The paper is organized in the following manner. We continue this introductory section with a review of the basic definitions and notations about HilbertC*-modules. Section 2 investigates some basic results of continuous g-frames in Hilbert C*-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 C*-modules.Let us recall the definitions and some basic properties of HilbertC*-modules. For more details, the interested readers can refer to the books by Lance [22] and Wegge-Olsen [23]. Let A be a C*-algebra with involution *. A pre-Hilbert C*-module over A or, simply, a pre-Hilbert A-module, is a complex linear space 𝒰 which is a left A-module with map 〈·,·〉:𝒰×𝒰→A, called an A-valued inner product, and it possesses the following properties:(1)
〈f,f〉≥0 for all f∈𝒰 and 〈f,f〉=0 if and only if f=0;(2)
〈f,g〉=〈g,f〉* for all f,g∈𝒰;(3)
〈af+g,h〉=a〈f,h〉+〈g,h〉 for all a∈A, f,g,h∈𝒰;(4)
〈λf,g〉=λ〈f,g〉 whenever λ∈ℂ and f,g∈𝒰.Forf∈𝒰, we define a norm on 𝒰 by ∥f∥𝒰=∥〈f,f〉∥A1/2. If 𝒰 is complete with this norm, it is called a Hilbert C*-module over A or a Hilbert A-module.Let(𝒰,〈·,·〉1) and (𝒱,〈·,·〉2) be two Hilbert A-modules. A map T:𝒰→𝒱 is said to be adjointable if there exists a map S:𝒱→𝒰 such that 〈Tf,g〉2=〈f,Sg〉1 for all f∈𝒰 and g∈𝒱. We denote by EndA*(𝒰,𝒱) the collection of all adjointable A-linear maps from 𝒰 to 𝒱. The following two lemmas will be used in the later section.Lemma 1 (see [24]).
Letℳ and 𝒩 be two Hilbert A-modules over a C*-algebra A and let T:ℳ→𝒩 be a linear map. Then the following conditions are equivalent:(1)
the operatorT is bounded and A-linear;(2)
there exists a constantK≥0 such that the inequality 〈Tx,Tx〉≤K〈x,x〉 holds in A for all x∈ℳ.Lemma 2 (see [25]).
LetA be a C*-algebra, let 𝒰 and 𝒱 be two Hilbert A-modules, and let T∈EndA*(𝒰,𝒱). 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∈𝒱;(3)
T* is bounded below with respect to inner product; that is, there is m′>0 such that 〈T*f,T*f〉≥m′〈f,f〉 for all f∈𝒱.Let𝒱 be a Hilbert A-module and {𝒱m}m∈ℳ a sequence of closed submodules of 𝒱. Set
(1)⨁m∈ℳ𝒱m={g={gm}:gm∈𝒱m,∥∫m∈ℳ‍〈gm,gm〉dμ(m)∥<∞}.For anyf={fm:m∈ℳ} and g={gm:m∈ℳ}, if the A-valued inner product is defined by 〈f,g〉=∫m∈ℳ‍〈fm,gm〉dμ(m) and the norm is defined by ∥f∥=∥〈f,f〉∥1/2, then ⨁m∈ℳ𝒱m is a Hilbert A-module (see [22]).Throughout this paper,A is a unital C*-algebra, 𝒰 and 𝒱 are Hilbert A-modules, and {𝒱m}m∈ℳ is a sequence of closed submodules of 𝒱. For T∈EndA*(𝒰,𝒱), we use Ran(T) and 𝒩(T) to denote the range and the null space of  T, respectively. As usual, we use I𝒰 to denote the identity operator on 𝒰.


## 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 HilbertC*-modules and, in particular, we obtain an equivalent condition under which a Hilbert C*-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 C*-modules.Definition 3 (see [21]).
We call a family of adjointableA-linear operators {Λm∈EndA*(𝒰,𝒱m):m∈ℳ} a continuous generalized frame or simply a continuous g-frame for Hilbert C*-module 𝒰 with respect to {𝒱m:m∈ℳ} if(1)
for anyf∈𝒰, the function f~:ℳ→𝒱m defined by f~(m)=Λmf is measurable;(2)
there is a pair of constantsA,B>0 such that, for any f∈𝒰,
(2)A〈f,f〉≤∫m∈ℳ‍〈Λmf,Λmf〉dμ(m)≤B〈f,f〉.The constants A and B are called continuous g-frame bounds. We call {Λm:m∈ℳ} a continuous tight g-frame if A=B and a continuous Parseval g-frame if A=B=1. If only the right-hand inequality of (2) is satisfied, we call {Λm:m∈ℳ} a continuous g-Bessel sequence for 𝒰 with respect to {𝒱m:m∈ℳ} with Bessel bound B.
We have the following equivalent definition for continuous g-Bessel sequences in HilbertC*-modules.Proposition 4.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} be a sequence of adjointable A-linear operators on 𝒰. Then {Λm:m∈ℳ} is a continuous g-Bessel sequence with Bessel bound D if and only if, for all f∈𝒰,
(3)∥∫m∈ℳ‍〈Λmf,Λmf〉dμ(m)∥≤D∥f∥2.Proof.
“⇒”. It is obvious.
“⇐”. Define a linear operatorT:𝒰→⨁m∈ℳ𝒱m by Tf={Λmf:m∈ℳ} for all f∈𝒰. Then
(4)∥Tf∥2=∥〈Tf,Tf〉∥=∥∫m∈ℳ‍〈Λmf,Λmf〉dμ(m)∥≤D∥f∥2,
which implies that ∥Tf∥≤D∥f∥. Hence, T is bounded. It is clear that T is A-linear. Then by Lemma 1, we have 〈Tf,Tf〉≤D〈f,f〉, equivalently, ∫m∈ℳ〈Λmf,Λmf〉dμ(m)≤D〈f,f〉, as desired.The following proposition gives an equivalent condition for a continuous g-Bessel sequence to be a continuous g-frame.Proposition 5.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} be a continuous g-Bessel sequence for 𝒰 with respect to {𝒱m:m∈ℳ}. Then {Λm:m∈ℳ} is a continuous g-frame for 𝒰 if and only if there exists a constant C>0 such that
(5)C∥f∥2≤∥∫m∈ℳ‍〈Λmf,Λmf〉dμ(m)∥,∀f∈𝒰.Proof.
“⇒”. It is straightforward.
“⇐”. We define a linear operator as follows:(6)T:𝒰⟶⨁m∈ℳ𝒱m,Tf={Λmf:m∈ℳ},∀f∈𝒰.
Then T is adjointable. Indeed,
(7)〈Tf,g〉=∫m∈ℳ‍〈Λmf,gm〉dμ(m)=〈f,∫m∈ℳ‍Λm*gmdμ(m)〉,
for all f∈𝒰,g={gm}∈⨁m∈ℳ𝒱m. It follows from (5) that
(8)∥Tf∥2=∥〈Tf,Tf〉∥=∥∫m∈ℳ‍〈Λmf,Λmf〉dμ(m)∥≥C∥f∥2.
Thus, ∥Tf∥≥C∥f∥ for all f∈𝒰. Then by Lemma 2, there exists M>0 such that 〈Tf,Tf〉≥M〈f,f〉; that is, M〈f,f〉≤∫m∈ℳ‍〈Λmf,Λmf〉dμ(m). 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 Lemma20.Proposition 6.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} and {Γm∈EndA*(𝒰,𝒱m):m∈ℳ} be two continuous g-Bessel sequences for 𝒰 with respect to {𝒱m:m∈ℳ}. If f=∫m∈ℳ‍Λm*Γmfdμ(m) holds for all f∈𝒰, then both {Λm:m∈ℳ} and {Γm:m∈ℳ} are continuous g-frames for 𝒰 with respect to {𝒱m:m∈ℳ}.Proof.
Let us denote the Bessel bound of{Γm:m∈ℳ} by D. For all f∈𝒰, we have
(9)∥f∥4=∥〈∫m∈ℳ‍Λm*Γmfdμ(m),f〉∥2=∥∫m∈ℳ‍〈Λmf,Γmf〉dμ(m)∥2≤∥∫m∈ℳ‍〈Λmf,Λmf〉dμ(m)∥×∥∫m∈ℳ‍〈Γmf,Γmf〉dμ(m)∥≤D∥f∥2∥∫m∈ℳ‍〈Λmf,Λmf〉dμ(m)∥.
It follows that
(10)D-1∥f∥2≤∥∫m∈ℳ‍〈Λmf,Λmf〉dμ(m)∥.
Similarly, we can show that {Γm:m∈ℳ} is a continuous g-frame for 𝒰.Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} be a continuous g-Bessel sequence for 𝒰 with respect to {𝒱m:m∈ℳ}, we define the synthesis operator TΛ:⨁m∈ℳ𝒱m→𝒰 by
(11)TΛg=∫m∈ℳ‍Λm*gmdμ(m),∀g={gm}∈⨁m∈ℳ𝒱m.
It follows immediately from the observation that for all f∈𝒰,g={gm}∈⨁m∈ℳ𝒱m, and
(12)〈TΛg,f〉=∫m∈ℳ‍〈gm,Λmf〉dμ(m)=〈g,{Λmf:m∈ℳ}〉,TΛ is adjointable and its adjoint operator TΛ*:𝒰→⨁m∈ℳ𝒱m is given by TΛ*f={Λmf:m∈ℳ} for all f∈𝒰. We call TΛ* the analysis operator. By composing TΛ and TΛ*, we obtain the frame operator SΛ:𝒰→𝒰. Note that SΛ is a positive, self-adjoint operator which is invertible if and only if {Λm:m∈ℳ} is a continuous g-frame of 𝒰. If {Λm:m∈ℳ} is a continuous g-frame, then every f∈𝒰 has a representation of the form
(13)f=∫m∈ℳ‍Λm*ΛmSΛ-1fdμ(m)=∫m∈ℳ‍SΛ-1Λm*Λmfdμ(m).We can characterize the continuous g-frames in HilbertC*-modules in terms of the associated synthesis and analysis operators.Proposition 7.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} be a family of adjointable A-linear operators on 𝒰. Then the following statements are equivalent:(1)
{Λm:m∈ℳ}is a continuous g-frame for𝒰with respect to{𝒱m:m∈ℳ};(2)
the synthesis operatorTΛis well defined and surjective;(3)
the analysis operatorTΛ*is bounded below with respect to norm.Proof.
(
1
)
⇔
(
2
). See [21, Theorem 4.3].
(
2
)
⇔
(
3
). It follows directly from Lemma 2.We are now ready to present a necessary and sufficient condition for a HilbertC*-module to have a continuous g-frame.Theorem 8.
A HilbertA-module 𝒰 has a continuous g-frame with respect to {𝒱m:m∈ℳ} if and only if there exists an adjointable and invertible map from 𝒰 to a closed submodule of ⨁m∈ℳ𝒱m.Proof.
“⇒”. Assume that{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} is a continuous g-frame for 𝒰 with respect to {𝒱m:m∈ℳ} with synthesis operator TΛ. It follows from Proposition 7 that the analysis operator TΛ* is bounded below with respect to norm; and, consequently, TΛ* is injective with closed range. Now, TΛ* is an adjointable and invertible map from 𝒰 to Ran(TΛ*), which is a closed submodule of ⨁m∈ℳ𝒱m.
“⇐”. Suppose thatM is a closed submodule of ⨁m∈ℳ𝒱m and S:𝒰→M is an adjointable and invertible map. We define a family of adjointable operators as follows:
(14)Pm:⨁m∈ℳ𝒱m⟶𝒱m,Pm({Fm})=Fm,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀{Fm}∈⨁m∈ℳ𝒱m.
Taking Λm=PmS for each m∈ℳ, then
(15)∫m∈ℳ〈Λmf,Λmf〉dμ(m)=∫m∈ℳ〈PmSf,PmSf〉dμ(m)=〈Sf,Sf〉=〈S*Sf,f〉.
Hence, by [22, Proposition 1.2], we have
(16)∥S-1∥-2〈f,f〉≤∫m∈ℳ〈Λmf,Λmf〉dμ(m)≤∥S∥2〈f,f〉.It is easy to see that a continuous g-Bessel sequence{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} for 𝒰 with respect to {𝒱m:m∈ℳ} is a continuous g-frame if and only if there exists a continuous g-Bessel sequence {Γm∈EndA*(𝒰,𝒱m):m∈ℳ} for 𝒰 with respect to {𝒱m:m∈ℳ} such that
(17)f=∫m∈ℳ‍Λm*Γmfdμ(m),∀f∈𝒰.
In this case, we call {Γm:m∈ℳ} a dual continuous g-frame of {Λm:m∈ℳ}. If SΛ is the frame operator of {Λm∈EndA*(𝒰,𝒱m):m∈ℳ}, a continuous g-frame for 𝒰 with respect to {𝒱m:m∈ℳ}, then, a direct calculation yields that {ΛmSΛ-1:m∈ℳ} is a dual continuous g-frame of {Λm:m∈ℳ}; 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{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} be a continuous g-frame for 𝒰 with respect to {𝒱m:m∈ℳ} which possesses more than one dual, and let SΛ be the frame operator for {Λm:m∈ℳ}. Then for any dual continuous g-frame {Γm:m∈ℳ} of {Λm:m∈ℳ}, the inequality
(18)∫m∈ℳ〈ΛmSΛ-1f,ΛmSΛ-1f〉dμ(m)≤∫m∈ℳ‍〈Γmf,Γmf〉dμ(m)
is valid for all f∈𝒰. Besides, the quality holds precisely if Γm=ΛmSΛ-1 for all m∈ℳ.
More generally, wheneverf=∫m∈ℳ‍Λm*gmdμ(m) for certain g={gm}∈⨁m∈ℳ𝒱m, we have
(19)∫m∈ℳ‍〈gm,gm〉dμ(m)=∫m∈ℳ‍〈ΛmSΛ-1f,ΛmSΛ-1f〉dμ(m)+∫m∈ℳ‍〈gm-ΛmSΛ-1f,gm-ΛmSΛ-1f〉dμ(m).Proof.
We begin with showing the first statement. Since{Γm:m∈ℳ} is a dual continuous g-frame of {Λm:m∈ℳ}, it follows that ∫m∈ℳ‍(Λm*Γmf-Λm*ΛmSΛ-1f)dμ(m)=0 for all f∈𝒰. Therefore,
(20)∫m∈ℳ‍〈Γmf,Γmf〉dμ(m)=∫m∈ℳ‍〈Γmf-ΛmSΛ-1f+ΛmSΛ-1f,Γmf-ΛmSΛ-1f+ΛmSΛ-1f〉dμ(m)=∫m∈ℳ‍〈ΛmSΛ-1f,ΛmSΛ-1f〉dμ(m)+∫m∈ℳ‍〈Γmf-ΛmSΛ-1f,Γmf-ΛmSΛ-1f〉dμ(m),
showing that the first part of the assertion holds since
(21)∫m∈ℳ‍〈Γmf-ΛmSΛ-1f,Γmf-ΛmSΛ-1f〉dμ(m)≥0.
Now, suppose thatf∈𝒰 has two decompositions
(22)f=∫m∈ℳ‍Λm*ΛmSΛ-1fdμ(m)=∫m∈ℳ‍Λm*gmdμ(m),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiig={gm}∈⨁m∈ℳ𝒱m.
Since
(23)∫m∈ℳ‍〈gm,ΛmSΛ-1f〉dμ(m)=〈f,SΛ-1f〉=∫m∈ℳ‍〈ΛmSΛ-1f,gm〉dμ(m),
it follows that
(24)∫m∈ℳ‍〈gm,gm〉dμ(m)=∫m∈ℳ‍〈ΛmSΛ-1f,ΛmSΛ-1f〉dμ(m)+∫m∈ℳ‍〈gm-ΛmSΛ-1f,gm-ΛmSΛ-1f〉dμ(m).Definition 10 (see [21]).
A continuous g-frame{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} for Hilbert C*-module 𝒰 with respect to {𝒱m:m∈ℳ} is said to be a continuous g-Riesz basis if it satisfies the following:(1)
Λm≠0 for any m∈ℳ;(2)
if∫m∈𝒦‍Λm*gmdμ(m)=0, then Λm*gm is equal to zero for each m∈𝒦, where {gm}m∈𝒦∈⨁m∈ℳ𝒱m and 𝒦 is a measurable subset of ℳ.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 adjointableA-linear operators {Λm∈EndA*(𝒰,𝒱m):m∈ℳ} is a continuous g-Riesz basis for 𝒰 with respect to {𝒱m:m∈ℳ} if and only if the synthesis operator TΛ is a homeomorphism.We note, however, that in the proof of the above theorem, they said that “Λm*fm=0 for any m∈ℳ and Λm≠0, so fm=0”, which is not true, because if Λm has a dense range, then Λm* 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{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} of adjointable A-linear operators on 𝒰 a modular continuous g-Riesz basis if(1)
{f∈𝒰:Λmf=0,m∈ℳ}={0};(2)
there exist constantsA,B>0 such that for any g={gm}∈⨁m∈ℳ𝒱m,
(25)A∥g∥2≤∥∫m∈ℳ‍Λm*gmdμ(m)∥2≤B∥g∥2.Theorem 13 (see [26]).
A sequence{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} is a modular continuous g-Riesz basis for 𝒰 with respect to {𝒱m:m∈ℳ} if and only if the synthesis operator TΛ is a homeomorphism.Proof.
Suppose first that{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} is a modular continuous g-Riesz basis for 𝒰 with synthesis operator TΛ. Then (25) turns to be
(26)A∥g∥2≤∥TΛg∥2≤B∥g∥2,∀g={gm}∈⨁m∈ℳ𝒱m,
showing that TΛ is bounded below with respect to norm. Hence, by Lemma 2, its adjoint operator TΛ* is surjective. Since condition (1) in Definition 12 implies that TΛ* is injective, it follows that TΛ* is invertible, and so TΛ is invertible.
Conversely, letTΛ be a homeomorphism. Then TΛ* is injective. So condition (1) in Definition 12 holds. Now, for any g={gm}∈⨁m∈ℳ𝒱m,
(27)∥TΛ-1∥-2∥g∥2≤∥∫m∈ℳ‍Λm*gmdμ(m)∥2=∥TΛg∥2≤∥TΛ∥2∥g∥2.
Therefore, {Λm∈EndA*(𝒰,𝒱m):m∈ℳ} is a modular continuous g-Riesz basis for 𝒰 with respect to {𝒱m:m∈ℳ}.The following is an immediate consequence of Theorem13.Corollary 14.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} be a continuous g-frame for 𝒰 with respect to {𝒱m:m∈ℳ} with synthesis operator TΛ, then it is a modular continuous g-Riesz basis for 𝒰 with respect to {𝒱m:m∈ℳ} if and only if TΛ* is surjective.Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} and {Γm∈EndA*(𝒰,𝒱m):m∈ℳ} be continuous g-Bessel sequences for 𝒰 with respect to {𝒱m:m∈ℳ}. In [21], the authors defined an adjointable operator L about them as follows:
(28)L:𝒰⟶𝒰,Lf=∫m∈ℳ‍Γm*Λmfdμ(m).Theorem 15.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} be a continuous g-frame for 𝒰 with respect to {𝒱m:m∈ℳ} with bounds A,B and frame operator SΛ, and {Γm∈EndA*(𝒰,𝒱m):m∈ℳ} is a continuous g-Bessel sequence for 𝒰 with respect to {𝒱m:m∈ℳ}. Suppose that there exists a number 0<λ<A such that for all f∈𝒰,
(29)∥Lf-SΛf∥≤λ∥f∥.
Then {Λm:m∈ℳ} is a modular continuous g-Riesz basis for 𝒰 if and only if {Γm:m∈ℳ} is a modular continuous g-Riesz basis for 𝒰.Proof.
For anyf∈𝒰, we have
(30)∥Lf∥=∥Lf-SΛf+SΛf∥≥∥SΛf∥-∥Lf-SΛf∥≥(A-λ)∥f∥.
So, L is bounded below with respect to norm. On the other hand, since
(31)∥L*f-SΛf∥≤∥L*-SΛ∥∥f∥=∥(L-SΛ)*∥∥f∥≤λ∥f∥,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀f∈𝒰,
by the above result, L* is also bounded below with respect to norm, and hence, by Lemma 2, both L and L* are surjective, and furthermore, L is invertible. Let TΛ and TΓ be the synthesis operators of {Λm:m∈ℳ} and {Γm:m∈ℳ}, respectively. It is easy to check that L=TΓTΛ*. Thus, TΛ* is invertible if and only if TΓ* is invertible, and consequently, {Λm:m∈ℳ} is a modular continuous g-Riesz basis for 𝒰 if and only if {Γm:m∈ℳ} is a modular continuous g-Riesz basis for 𝒰.


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

The definitions of similar and unitary equivalent frames give rise to definitions of similar and unitary equivalent continuous g-frames in HilbertC*-modules.Definition 16.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} and {Γm∈EndA*(𝒰,𝒱m):m∈ℳ} be two continuous g-frames for 𝒰 with respect to {𝒱m:m∈ℳ}. One has the following.(1)
They are said to be similar or equivalent if there is an adjointable and invertible operatorT:𝒰→𝒰 such that Γm=ΛmT for each m∈ℳ.(2)
They are said to be unitary equivalent if there exists an adjointable and unitary linear operatorU:𝒰→𝒰 such that Γm=ΛmU for each m∈ℳ.Theorem 17.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} and {Γm∈EndA*(𝒰,𝒱m):m∈ℳ} be two continuous g-frames for 𝒰 with respect to {𝒱m:m∈ℳ} with synthesis operators TΛ and TΓ, respectively. Then the following statements are equivalent:(1)
there is an adjointable and invertible operatorT:𝒰→𝒰 such that Γm=ΛmT* for each m∈ℳ; that is, {Λm:m∈ℳ} and {Γm:m∈ℳ} are similar;(2)
there exists a constantM>0 such that
(32)〈(TΛ-TΓ)g,(TΛ-TΓ)g〉≤M·min{〈TΛg,TΛg〉,〈TΓg,TΓg〉}for all g={gm}∈⨁m∈ℳ𝒱m. Moreover, if (2) holds, then
(33)1(1+M)2〈f,f〉≤〈Tf,Tf〉≤(1+M)2〈f,f〉,∀f∈𝒰.Proof.
(
1
)
⇒
(
2
). Suppose that T:𝒰→𝒰 is an adjointable and invertible operator such that Γm=ΛmT* for each m∈ℳ. If f=∫m∈ℳ‍Λm*gmdμ(m) for certain g={gm}∈⨁m∈ℳ𝒱m, then we have
(34)Tf=∫m∈ℳ‍TΛm*gmdμ(m)=∫m∈ℳ‍(ΛmT*)*gmdμ(m)=∫m∈ℳ‍Γm*gmdμ(m).
Therefore,
(35)〈f-Tf,f-Tf〉=〈f,f〉+〈Tf,Tf〉+〈(-T*-T)f,f〉≤〈f,f〉+∥T∥2〈f,f〉+∥T*+T∥〈f,f〉≤(1+∥T∥)2〈f,f〉.
On the other hand,(36)〈f-Tf,f-Tf〉=〈f,f〉+〈Tf,Tf〉+〈(-T*-T)f,f〉=〈T-1Tf,T-1Tf〉+〈Tf,Tf〉+〈(-T*-T)T-1Tf,T-1Tf〉≤(∥T-1∥2+1)〈Tf,Tf〉+〈(T-1)*(-T*-T)T-1Tf,Tf〉≤(∥T-1∥2+1)〈Tf,Tf〉+∥(T-1)*(-T*-T)T-1∥〈Tf,Tf〉≤(1+∥T-1∥)2〈Tf,Tf〉.
Hence, (32) follows.
(
2
)
⇒
(
1
). For each f=∫m∈ℳ‍Λm*gmdμ(m)∈𝒰, we define an operator T:𝒰→𝒰 as follows:
(37)Tf=T(∫m∈ℳ‍Λm*gmdμ(m))=∫m∈ℳ‍Γm*gmdμ(m).
It is clear that T is well defined, and furthermore, T is adjointable. A simple calculation shows that its adjoint operator T* is given by
(38)T*h=∫m∈ℳ‍SΛ-1Λm*Γmhdμ(m),∀h∈𝒰,
where SΛ is the frame operator of {Λm:m∈ℳ}. Since TΓ is surjective by Proposition 7, it follows that T is also surjective. And (32) implies that T is injective, and so T is invertible. It remains to establish that Γm=ΛmT* for each m∈ℳ. For all f∈𝒰,g={gm}∈⨁m∈ℳ𝒱m, we have
(39)∫m∈ℳ‍〈gm,ΛmT*f〉dμ(m)=〈∫m∈ℳ‍TΛm*gmdμ(m),f〉=〈T∫m∈ℳ‍Λm*gmdμ(m),f〉=〈∫m∈ℳ‍Γm*gmdμ(m),f〉=∫m∈ℳ‍〈gm,Γmf〉dμ(m).
That is, 〈g,{ΛmT*f-Γmf:m∈ℳ}〉=0. Hence, Γm=ΛmT* for each m∈ℳ.
For the last statement, the assumptions implies that∥f-Tf∥≤M∥f∥ and ∥f-Tf∥≤M∥Tf∥ for all f∈𝒰. If we replace f by T-1f in the last inequality, we have ∥(T-1-I𝒰)f∥≤M∥f∥. Therefore,
(40)〈f,f〉=〈f-Tf,f-Tf〉+〈Tf,Tf〉+(〈f-Tf,Tf〉+〈Tf,f-Tf〉)≤(M+1)〈Tf,Tf〉+〈(T-1)*(T*(I𝒰-T)+(I𝒰-T)*T)×T-1Tf,Tf(T-1)*(T*(I𝒰-T)+(I𝒰-T)*T)〉≤(M+1)〈Tf,Tf〉+∥(T-1)*(T*(I𝒰-T)+(I𝒰-T)*T)T-1∥×〈Tf,Tf〉≤(1+M+2∥T-1-I𝒰∥)〈Tf,Tf〉≤(1+M)2〈Tf,Tf〉,〈Tf,Tf〉=〈Tf-f,Tf-f〉+〈f,f〉+〈Tf-f,f〉+〈f,Tf-f〉≤(M+1)〈f,f〉+〈((T-I𝒰)+(T-I𝒰)*)f,f〉≤((M+1)+∥(T-I𝒰)+(T-I𝒰)*∥)×〈f,f〉≤(1+M+2∥T-I𝒰∥)〈f,f〉≤(1+M)2〈f,f〉.
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 C*-modules.Proposition 18.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} and {Γm∈EndA*(𝒰,𝒱m):m∈ℳ} be two continuous Parseval g-frames for 𝒰 with respect to {𝒱m:m∈ℳ} with synthesis operators TΛ and TΓ, respectively. Then(1)
Ran(TΓ*)⊆Ran(TΛ*)if and only if there exists an adjointable operatorU:𝒰→𝒰which preserves inner product such thatΓm=ΛmUfor eachm∈ℳ. Conversely, ifU:𝒰→𝒰is an adjointable operator which preserves inner product such thatΓm=ΛmUfor eachm∈ℳ, then(41)Ran(TΛ*)=TΛ*(𝒩(U*))⊕Ran(TΓ*);(2)
Ran(TΓ*)=Ran(TΛ*)if and only if{Λm:m∈ℳ}and{Γm:m∈ℳ}are unitary equivalent.Proof.
(1) “⇒”. Assume thatRan(TΓ*)⊆Ran(TΛ*). Let us denote P=TΛ*TΛ and Q=TΓ*TΓ. Since both TΛ and TΓ are surjective, we know that Ran(P)=Ran(TΛ*) and Ran(Q)=Ran(TΓ*). Since {Λm:m∈ℳ} and {Γm:m∈ℳ} are two continuous Parseval g-frames for 𝒰, it follows that P and Q are orthogonal projections from ⨁m∈ℳ𝒱m onto Ran(TΛ*) and Ran(TΓ*), respectively. Let U=TΛTΓ*, then, for an arbitrary element f of 𝒰, recalling that TΓ*f∈Ran(TΛ*), we have
(42)U*Uf=TΓTΛ*TΛTΓ*f=TΓTΓ*f=f.
Thus, U preserves inner product. Also,
(43)U*f=TΓTΛ*f=∫m∈ℳ‍Γm*Λmfdμ(m),
and so,
(44)∫m∈ℳ‍〈ΛmUf,Γmf〉dμ(m)=〈∫m∈ℳ‍Γm*ΛmUfdμ(m),f〉=〈U*Uf,f〉=〈f,f〉.
Note that
(45)∫m∈ℳ‍〈Γmf,Γmf〉dμ(m)=〈f,f〉,∫m∈ℳ‍〈ΛmUf,ΛmUf〉dμ(m)=〈Uf,Uf〉=〈f,f〉;
it follows that
(46)〈{(Γm-ΛmU)f:m∈ℳ},{(Γm-ΛmU)f:m∈ℳ}〉=∫m∈ℳ‍〈(Γm-ΛmU)f,(Γm-ΛmU)f〉dμ(m)=∫m∈ℳ‍〈Γmf,Γmf〉dμ(m)-∫m∈ℳ‍〈Γmf,ΛmUf〉dμ(m)-∫m∈ℳ‍〈ΛmUf,Γmf〉dμ(m)+∫m∈ℳ‍〈ΛmUf,ΛmUf〉dμ(m)=〈f,f〉-〈f,f〉-〈f,f〉+〈f,f〉=0.
Hence, {(Γm-ΛmU)f:m∈ℳ}=0 for each f∈𝒰, and Γm=ΛmU for each m∈ℳ as a consequence.
“⇐”. It is obvious.
For the second part of (1), sinceTΛ* is an isometry, it follows that
(47)Ran(TΛ*)=TΛ*(𝒩(U*)⊕(𝒩(U*))⊥)=TΛ*(𝒩(U*)⊕Ran(U))=TΛ*(𝒩(U*))⊕Ran(TΛ*U)=TΛ*(𝒩(U*))⊕Ran(TΓ*).
(2) Suppose thatRan(TΓ*)=Ran(TΛ*), then (41) implies that TΛ*(𝒩(U*))=0, and hence, 𝒩(U*)=0. Thus, U* is injective, and so, U is invertible. Since U*U=I𝒰, it follows that U is unitary. For the other implication, let U:𝒰→𝒰 be a unitary linear operator such that Γm=ΛmU for each m∈ℳ. Then TΓ*=TΛ*U, and so, Ran(TΓ*)=Ran(TΛ*).For the general case, we have the following proposition.Proposition 19.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} and {Γm∈EndA*(𝒰,𝒱m):m∈ℳ} be two continuous g-frames for 𝒰 with respect to {𝒱m:m∈ℳ} with synthesis operators TΛ and TΓ and frame operators SΛ and SΓ, respectively. Then(1)
Ran(TΓ*)⊆Ran(TΛ*)if and only if there exists an adjointable operatorU:𝒰→𝒰such thatΓm=ΛmUfor eachm∈ℳ;(2)
Ran(TΓ*)=Ran(TΛ*)if and only if{Λm:m∈ℳ}and{Γm:m∈ℳ}are similar.Proof.
(1) “⇒”. Assume thatRan(TΓ*)⊆Ran(TΛ*). We already know that Ran(TΛ*) and Ran(TΓ*) are closed submodules of ⨁m∈ℳ𝒱m. Then Ran(TΛ*)=(𝒩(TΛ))⊥ and Ran(TΓ*)=(𝒩(TΓ))⊥, and thus, 𝒩(TΛ)⊆𝒩(TΓ). It is easy to check that Λ′={ΛmSΛ-1/2:m∈ℳ} and Γ′={ΓmSΓ-1/2:m∈ℳ} are both continuous Parseval g-frames. Let us denote by TΛ′ and TΓ′ the synthesis operators of Λ′ and Γ′, respectively. Then TΛ′=SΛ-1/2TΛ and TΓ′=SΓ-1/2TΓ. Therefore, 𝒩(TΛ′)=𝒩(TΛ) and 𝒩(TΓ′)=𝒩(TΓ). By Proposition 18, there exists an adjointable operator S:𝒰→𝒰 such that ΓmSΓ-1/2=ΛmSΛ-1/2S for each m∈ℳ. Hence, the result follows by letting U=SΛ-1/2SSΓ1/2.
“⇐”. It is straightforward.
(2) “⇒”. IfRan(TΓ*)=Ran(TΛ*), then Ran(TΓ′*)=Ran(TΛ′*). By part (2) of Proposition 18, S is unitary, and consequently, U=SΛ-1/2SSΓ1/2 is invertible.
“⇐”. It is obvious.


## 4. Stability of Duals of Continuous g-Frames in HilbertC*-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 C*-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{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} be a continuous g-frame for 𝒰 with respect to {𝒱m:m∈ℳ} with bounds A,B and the synthesis operator TΛ. Then there exists a one-to-one correspondence between the duals of {Λm:m∈ℳ} and operator ψ∈EndA*(𝒰,⨁m∈ℳ𝒱m) such that TΛψ=0.Proof.
Assume first that{Γm∈EndA*(𝒰,𝒱m):m∈ℳ} is a dual continuous g-frame of {Λm:m∈ℳ} with bounds A1 and B1, and let SΛ be the frame operator of {Λm:m∈ℳ}. Define ψ:𝒰→⨁m∈ℳ𝒱m,f↦ψf by
(48)(ψf)m=Γmf-ΛmSΛ-1f,m∈ℳ.
Then ψ is adjointable, that is; ψ∈EndA*(𝒰,⨁m∈ℳ𝒱m). Indeed,
(49)〈ψf,g〉=∫m∈ℳ‍〈(ψf)m,gm〉dμ(m)=∫m∈ℳ‍〈Γmf-ΛmSΛ-1f,gm〉dμ(m)=∫m∈ℳ‍〈f,Γm*gm〉dμ(m)-∫m∈ℳ‍〈f,SΛ-1Λm*gm〉dμ(m)=∫m∈ℳ‍〈f,Γm*gm-SΛ-1Λm*gm〉dμ(m)=〈f,∫m∈ℳ‍(Γm*gm-SΛ-1Λm*gm)dμ(m)〉,
for all f∈𝒰,g={gm}∈⨁m∈ℳ𝒱m. Moreover, we have
(50)TΛψf=∫m∈ℳ‍Λm*(ψf)mdμ(m)=∫m∈ℳ‍Λm*(Γmf-ΛmSΛ-1f)dμ(m)=∫m∈ℳ‍Λm*Γmfdμ(m)-∫m∈ℳ‍Λm*ΛmSΛ-1fdμ(m)=f-f=0.
Conversely, letψ∈EndA*(𝒰,⨁m∈ℳ𝒱m) and TΛψ=0. Take
(51)Γmf=(ψf)m+ΛmSΛ-1f,f∈𝒰,m∈ℳ.
Since
(52)∥∫m∈ℳ‍〈Γmf,Γmf〉dμ(m)∥1/2=∥{Γmf}m∈ℳ∥≤∥{(ψf)m}m∈ℳ∥+∥{ΛmSΛ-1f}m∈ℳ∥≤∥ψf∥+∥∫m∈ℳ‍〈ΛmSΛ-1f,ΛmSΛ-1f〉dμ(m)∥1/2≤∥ψ∥∥f∥+1A∥f∥,
it follows that {Γm:m∈ℳ} is a continuous g-Bessel sequence for 𝒰 with respect to {𝒱m:m∈ℳ}. Furthermore,
(53)∫m∈ℳ‍Λm*Γmfdμ(m)=∫m∈ℳ‍Λm*(ψf)mdμ(m)+∫m∈ℳ‍Λm*ΛmSΛ-1fdμ(m)=TΛψf+f=f.
Thus, {Γm:m∈ℳ} is a dual continuous g-frame of {Λm:m∈ℳ}, by Proposition 6.Theorem 21.
Let{Λm∈EndA*(𝒰,𝒱m):m∈ℳ} and {Γm∈EndA*(𝒰,𝒱m):m∈ℳ} be two continuous g-frames for 𝒰 with respect to {𝒱m:m∈ℳ} with bounds A1,B1 and A2,B2, respectively. Also, let {Λm′:m∈ℳ} be a fixed dual of {Λm:m∈ℳ} with frame bounds A3,B3. If {Λm-Γm:m∈ℳ} is a continuous g-Bessel sequence with Bessel bound ϵ, then there exists a dual {Γm′:m∈ℳ} of {Γm:m∈ℳ} such that {Λm′-Γm′:m∈ℳ} is also a continuous g-Bessel sequence.Proof.
Let us denote byTΛ, TΓ and SΛ, SΓ the synthesis operators and frame operators of {Λm:m∈ℳ} and {Γm:m∈ℳ}, respectively. By the proof of Lemma 20 we know that there exists ψ∈EndA*(𝒰,⨁m∈ℳ𝒱m) with
(54)(ψf)m=12ϵ1(B3+1/A1)(Λm′f-ΛmSΛ-1f)
such that TΛψ=0 for all f∈𝒰,m∈ℳ. A simple calculation shows that
(55)∥ψ∥≤12ϵ1(B3+1/A1)(B3+1A1)=12ϵ.
Let(56)Θmf=ΓmSΓ-1f+(ψf)m,∀f∈𝒰,m∈ℳ.
It is easy to see that {Θm:m∈ℳ} is a continuous g-Bessel sequence. Let TΘ be the synthesis operator of {Θm:m∈ℳ}, then ∥TΘ∥=∥TΘ*∥≤1/A2+1/(2ϵ) and
(57)∥f-TΓTΘ*f∥=∥f-∫m∈ℳ‍Γm*Θmfdμ(m)∥=∥f-∫m∈ℳ‍Γm*ΓmSΓ-1fdμ(m)-∫m∈ℳ‍Γm*(ψf)mdμ(m)∥=∥TΓψf∥=∥TΓψf-TΛψf∥≤∥TΓ-TΛ∥∥ψ∥∥f∥≤ϵ∥ψ∥∥f∥≤ϵ12ϵ∥f∥=12∥f∥.
Hence, ∥I𝒰-TΓTΘ*∥<1, and furthermore, TΓTΘ* is invertible. Therefore, every f∈𝒰 can be represented by
(58)f=TΓTΘ*(TΓTΘ*)-1f=∫m∈ℳ‍Γm*Θm(TΓTΘ*)-1fdμ(m),
showing that {Γm′:m∈ℳ}={Θm(TΓTΘ*)-1:m∈ℳ} is a dual of {Γm:m∈ℳ}. In what follows, we will show that {Γm′:m∈ℳ} is the desired continuous g-frame.
If we takeT=(TΓTΘ*)-1, then
(59)∥T∥≤11-∥I𝒰-T-1∥≤11-ϵ∥ψ∥,
and so,
(60)∥I𝒰-T∥≤∥T∥∥I𝒰-T-1∥≤ϵ∥ψ∥1-ϵ∥ψ∥.
Denoting by TΛ′ the synthesis operator of {Λm′:m∈ℳ}, we have
(61)∥∫m∈ℳ‍(Λm′)*gmdμ(m)-∫m∈ℳ‍(Γm′)*gmdμ(m)∥=∥∫m∈ℳ‍(Λm′)*gmdμ(m)-∫m∈ℳ‍T*Θm*gmdμ(m)∥=∥TΛ′g-T*TΘg∥=∥TΛ′g-T*TΛ′g+T*TΛ′g-T*TΘg∥≤∥I𝒰-T∥∥TΛ′g∥+∥T∥∥TΛ′g-TΘg∥≤(∥I𝒰-T∥+∥T∥)∥TΛ′g∥+∥T∥∥TΘg∥≤1+ϵ∥ψ∥1-ϵ∥ψ∥B′∥g∥+11-ϵ∥ψ∥(1A2+12ϵ)∥g∥=(1+ϵ∥ψ∥)B1+1/A2+1/2ϵ1-ϵ∥ψ∥∥g∥,
where B′ is the upper frame bound of {Λm′:m∈ℳ}. The proof is completed.



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*Source: 192732-2013-10-02.xml*