# The Characterization and Stability of g-Riesz Frames for Super Hilbert Space

Authors: Dingli Hua; Yongdong Huang
Journal: Journal of Function Spaces
 (2015)

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## Abstract

G-frames and g-Riesz frames as generalized frames in Hilbert spaces have been studied by many authors in recent years. The super Hilbert space has a certain advantage compared with the Hilbert space in the field of studying quantum mechanics. In this paper, for super Hilbert spaceH⊕K, the definitions of a g-Riesz frame and minimal g-complete are put forward; also a characterization of g-Riesz frames is obtained. In particular, we generalize them to general super Hilbert space L1⊕L2⊕⋯⊕Ln. Finally, a conclusion of the stability  of a g-Riesz frame for the super Hilbert space is given.


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## Body

## 1. Introduction

Frame as generalized basis in Hilbert space was first introduced by Duffin and Schaeffer [1] during the studied nonharmonic Fourier series in 1952. In 1986, Daubechies et al. [2] reintroduced the concept of frame. Now the theory of frames has been widely used in many areas such as the characterization of function spaces, signal processing, filter theory, image processing and quantum mechanics. We refer to [3–10] for an introduction to frame theory in Hilbert space and its application.Sun [11] introduced g-Riesz basis and g-frame; g-frame actually generalized the concept of frame. Since then, g-frame, g-frame sequence, Besselian g-frame, near g-Riesz bases, and so on are focused on and studied by many authors. The authors [12] introduced Besselian g-frames and near g-Riesz bases in Hilbert space and gave some characterizations of them. In [13], g-Riesz frames were studied and some corresponding results were given. In [14], the concept of g-bases in Hilbert spaces was introduced and some properties about g-bases were proved. Because super Hilbert spaces arose naturally as the state space of a quantum field in the functional Schrödinger representation of spinor quantum field theory and it provided a means to bring supersymmetric quantum field theories into a form resembling standard quantum mechanics, the super Hilbert space has certain advantages compared with the Hilbert space in quantum mechanics. With the extensive research of super Hilbert space [15–20], scholars were beginning to study g-frames for super Hilbert spaces [20, 21]. Unfortunately, although g-Riesz frames were considered as a class of important frames, we have not consulted the literature of the g-Riesz frame for super Hilbert space H⊕K so far. Because g-Riesz frames play an important role in approximate calculating coefficients of g-frames, therefore, the study of g-Riesz frames for super Hilbert space H⊕K has a double meaning of theory and application. In order to enrich the frame theory, we give the concept of g-Riesz frame for super Hilbert space H⊕K and the characterization and necessary condition for g-Riesz frame. We also expand corresponding conclusions to general super Hilbert space L1⊕L2⊕⋯⊕Ln. Finally, we consider the stability of a g-Riesz frame for super Hilbert space.Throughout this paper,H and K are two complex Hilbert spaces and Hj:j∈J is a sequence of closed subspaces of H. LH,Hj is the collection of all bounded linear operators from H into Hj, where J is a subset of integers Z. l2Hjj∈J is defined by (1)l2Hjj∈J=ajj∈J:aj∈Hj,j∈J,∑j∈Jaj2<+∞,with the inner product given by (2)ajj∈J,bjj∈J=∑j∈Jaj,bjHj,and it is clear that l2Hjj∈J is a complex Hilbert space.The literature [16] gave the definition of super Hilbert space.Definition 1 (see [16, p.557]).
Super Hilbert space is a direct sum thatH=H0⊕H1 of two complex Hilbert spaces H0,·,·0,H1,·,·1 equipped with the super Hermitian form ·,·=·,·0+·,·1.


## 2. Preliminaries

In this section, some necessary definitions and lemmas are introduced.Definition 2 (see [11, Definition  1.1]).
A sequenceΛj∈LH,Hj:j∈J is called a g-frame for H with respect to Hjj∈J if there exist two positive constants A and B such that, for all f∈H, (3)Af2≤∑j∈JΛjf2≤Bf2.The constants A and B are called the lower and upper bounds of g-frame, respectively. If the right hand inequality holds, then we say that Λjj∈J is a g-Bessel sequence for H with respect to Hjj∈J. If A=B, we call this g-frame a tight g-frame. If a g-frame ceases to be a g-frame whenever any single element is removed from Λjj∈J, it is called an exact g-frame.Definition 3 (see [14, Definition  2.2]).
One says thatΛj∈LH,Hj:j∈J is g-complete, if f∈H:Λjf=0,forallj=0.Definition 4 (see [11, Definition  3.1]).
A sequenceΛj∈LH,Hj:j∈J is called a g-Riesz basis for H with respect to Hjj∈J, if the sequence Λjj∈J is g-complete and there exist positive constants A and B such that (4)A∑j∈J1gj2≤∑j∈J1Λj∗gj2≤B∑j∈J1gj2for all finite subset J1⊂J and gj∈Hj, j∈J1. The constants A and B are called the lower and upper bounds of g-Riesz bases, respectively.Definition 5 (see [11, Definition  3.1]).
LetΛj∈LH,Hj:j∈J. Suppose that ejkk∈Kj is an orthonormal basis for Hj, where Kj is a subset of Z. Denote (5)ujk=Λj∗ejk,j∈J,k∈Kj,(6)Λjf=∑k∈Kjf,ujkejk,∀f∈H.We call ujkj∈J,k∈Kj the sequence induced by Λjj∈J with respect to ejkj∈J,k∈Kj.Theorem 6 (see [21, Proposition  2.8]).
LetΛjj∈J and Γjj∈J be sequences in LH,Hj and LK,Hj, respectively, and let ejkk∈Kj be an orthonormal basis for Hj, where Kj is a subset of Z, and let ψjk=Λj∗ejk, φjk=Γj∗ejk, and Θjf,g=Λjf+Γjg. Then ψjk,φjkj∈J,k∈Kj is a frame (resp., Bessel sequence, Riesz basis) for super Hilbert space H⊕K if and only if Θj∈LH⊕K,Hj):j∈J is a g-frame (resp., g-Bessel sequence, g-Riesz basis) for H⊕K with respect to Hj.Proposition 7 (see [21, Proposition  2.9]).
LetΘj∈LH⊕K,Hj:j∈J, {(Λj∈L(H,Hj):j∈J}, and Γj∈LK,Hj:j∈J be g-frames, where Θjf,g=Λjf+Γjg. Then g-frame operator for Θj∈LH⊕K,Hj:j∈J is defined by (7)SΘf,g=∑j∈JΛj∗Λjf+Λj∗Γjg,∑j∈JΓj∗Λjf+Γj∗Γjg.The literature [21] introduced the concept of disjoint g-frames. A pair of g-frames Λj∈LH,Hj:j∈J and Γj∈LK,Hj:j∈J is called disjoint if Θjf,g is a g-frame for H⊕K, where Θjf,g=Λjf+Γjg.With the definition of g-complete of g-frame for Hilbert space, we give the definition of g-complete of g-frame for super Hilbert space as follows.Definition 8.
Θ
j
∈
L
H
⊕
K
,
H
j
j
=
1
∞ is called g-complete with respect to Hjj=1∞ under the condition of that f,g:Θjf,g=0,forallj=0,0.


## 3. Characterization of g-Riesz Frame for Super Hilbert Space

In this section, we first give the concept and the characterization of g-Riesz frame for super Hilbert spaceH⊕K, and then we generalize them to super Hilbert space L1⊕L2⊕⋯⊕Ln.


### 3.1. Characterization of g-Riesz Frame for Super Hilbert SpaceH⊕K

Before giving the characterization of g-Riesz frames for super Hilbert spaceH⊕K, we give the definition of g-Riesz frames and some related lemmas.Suppose thatI is a subset of J, and denote (8)H⊕KI=∑j∈I1Θj∗gj=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj:for any finiteI1⊂I,gj∈Hj,j∈I1¯.Definition 9.
Suppose thatΘj∈LH⊕K,Hj:j∈J is a g-frame for H⊕K with respect to Hjj∈J. One says that Θjj∈J is a g-Riesz frame if every subfamily Θjj∈I of Θjj∈J is a g-frame for H⊕KI with respect to Hjj∈I with uniform g-frame lower bounds.For the aboveH⊕KI, we have the following.Lemma 10.
Suppose that for everyj∈J, Λj∈LH,Hj, Γj∈LK,Hj, Θj∈L(H⊕K,Hj), and ejkk∈Kj is an orthonormal basis for Hj. ψjk=Λj∗ejk and φjm=Γj∗ejm are defined as in (5). Then H⊕KI=span¯ψjk,φjmj∈I,k,m∈Kj.Proof.
Denote(9)H⊕KI^=∑j∈I1Θj∗gj=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj:for any finiteI1⊂I,gj∈Hj,j∈I1.Since ψjk=Λj∗ejk, φjm=Γj∗ejm, j∈I, k,m∈Kj, we have ψjk,φjm∈H⊕KI^, j∈I, k,m∈Kj. This implies that spanψjk,φjmj∈I,k,m∈Kj⊂H⊕KI^. Therefore, span¯ψjk,φjmj∈I,k,m∈Kj⊂H⊕KI.
On the other hand, suppose thatf,g∈H⊕KI, and then there exists a finite subset I1⊂I and gj∈Hj, j∈I1, such that f,g=∑j∈I1Θj∗gj=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj. For every j∈I1, let gj=∑k∈Kj‍cjkejk, where cjkk∈Kj∈l2. Then, we have (10)Θj∗gj=Λj∗∑k∈Kjcjkejk,Γj∗∑k∈Kjcjkejk=∑k∈Kjcjkψjk,∑k∈Kjcjkφjk∈span¯⁡ψjk,φjmj∈I,k,m∈Kj.
Therefore,(11)f,g=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj∈span¯⁡ψjk,φjmj∈I,k,m∈Kj.This implies that H⊕KI⊂span¯ψjk,φjmj∈I,k,m∈Kj. The proof of Lemma 10 is completed.Lemma 11.
LetW be a closed subspace of H⊕K and {Θj∈L(H⊕K,Hj)}j∈J. Suppose that ψjk,φjm∈W for every j∈I, k,m∈Kj, where I is a subset of J and ψjkj∈J,k∈Kj, φjmj∈J,m∈Kj are defined as in (5). If Θjj∈I is g-complete in W, then {(ψjk,φjm)}j∈I,k,m∈Kj is complete in W.Proof.
For anyj∈I, k,m∈Kj, we have ψjk,φjm∈W, where W is a closed subspace of H⊕K. Furthermore, we obtain (12)span¯⁡ψjk,φjmj∈I,k,m∈Kj⊂W.It is enough to prove that if f,g∈W and f,g⊥span¯ψjk,φjmj∈I,k,m∈Kj, then f,g,ψjk,φjm=0 for j∈I, k,m∈Kj. By equality (6), we obtain (13)Θjf,g=Λjf+Γjg=∑k∈Kjf,ψjkejk+∑k∈Kjg,φjkejk=0,∀j∈I.Since Θjj∈I is g-complete in W, we have f,g=0,0. Hence, ψjk,φjmj∈J,k,m∈Kj is complete in W.Inspired by the concept of minimal g-complete of g-frame for Hilbert space, we give the definition of minimal g-complete of g-frame for super Hilbert space.Definition 12.
LetW be a closed subspace of H⊕K and I be a subset of J. If Θjj∈I is g-complete in W, but Θjj∈I∖j0 is not g-complete in W for any j0∈I, then one says that Θjj∈I is minimal g-complete in W.Lemma 13.
Suppose thatΘj∈LH⊕K,Hj for j∈J and I is any finite nonempty subset of J. Then there exists a finite nonempty subset I0⊂I such that Θjj∈I0 is minimal g-complete in H⊕KI0 and H⊕KI0=H⊕KI, where H⊕KI is defined as in (8).Proof.
We prove Lemma13 in two cases.
Case  1. Θjj∈I is g-complete in H⊕KI for any j0∈I, but Θjj∈I∖j0 is not g-complete in H⊕KI. Let I0=I. Then the conclusion is right.
Case  2. Suppose that there exists j0∈I such that Θjj∈I∖j0 is g-complete in H⊕KI. Let I1=I∖j0. If there still exists j1∈I1 such that ψjk,φjmj∈I1∖j1,k,m∈Kj is complete in span¯ψjk,φjmj∈I,k,m∈Kj, then we remove j2 from I1 in the same way. Repeat the operation above. Because I is nonempty and finite, this process must stop after finite steps. Assume that we remove j0,j1,…,jn from I, where n∈N and I0=I∖j0,j1,…,jn. Then, we obtain that I0⊂I satisfies the following two statements:
(
1
)
Θ
j
j
∈
I
0 is g-complete in H⊕KI.
(
2
) For any i0∈I0, Θjj∈I0∖i0 is not g-complete in H⊕KI.
By Lemmas10 and 11, the statement (1) implies that
(
3
)
ψ
j
k
,
φ
j
m
j
∈
I
0
,
k
,
m
∈
K
j is complete in span¯ψjk,φjmj∈I,k,m∈Kj.
Obviously,I0 is nonempty. The proof is by contradiction.
SupposeI0=∅, by (1), we obtain that Θjj∈I0 is g-complete in H⊕KI. It is obvious that this is impossible. So I0 is nonempty. Now we prove H⊕KI0=H⊕KI. By Definition 12 and Lemma 10, Θjj∈I is minimal g-complete in span¯ψjk,φjmj∈I,k,m∈Kj. By (3), we get (14)span¯⁡ψjk,φjmj∈I0,k,m∈Kj=span¯⁡ψjk,φjmj∈I,k,m∈Kj.By Lemma 10, we get H⊕KI0=H⊕KI. The proof of Lemma 13 is completed.Based on this, we can obtain a characterization of g-Riesz frame for super Hilbert spaceH⊕K.Theorem 14.
LetΘj∈LH⊕K,Hj:j∈J be a g-frame for H⊕K with respect to Hjj∈J. Then the following two statements are equivalent.
(1)Θjj∈J is a g-Riesz frame for H⊕K with respect to Hjj∈J.
(2) There existsA>0 such that Θjj∈I is minimal g-complete in H⊕KI for any nonempty subset I of J. And (15)Af,g2≤∑j∈IΘjf,g2,∀f,g∈H⊕KI,where H⊕KI is defined as in (8).Proof.
(
1
)
⇒
(
2
). Since Θjj∈J is a g-Riesz frame for H⊕K with respect to Hjj∈J, there exists A>0 such that Θjj∈I is a g-frame for H⊕KI with respect to Hjj∈I for any nonempty I of J. Then, we obtain (16)Af,g2≤∑j∈IΘjf,g2,∀f,g∈H⊕KI.
(
2
)
⇒
(
1
). Suppose that Θjj∈J is a g-frame for H⊕K with respect to Hjj∈J with upper bound B. Then (17)∑j∈JΘjf,g2≤Bf,g2,∀f,g∈H⊕K.
There are two cases to prove thatΘjj∈I is a g-frame for H⊕KI with respect to Hjj∈I, where I is any nonempty subset of J.
Case  1. When I is any finite nonempty subset of J, by Lemma 13, there exists a finite nonempty subset I0⊂I such that Θjj∈I0 is minimal g-complete in H⊕KI0 and H⊕KI0=H⊕KI. By (15), we have (18)Af,g2≤∑j∈I0Θjf,g2,∀f,g∈H⊕KI0=H⊕KI.
Again by (15), for ∀f,g∈H⊕KI, we have (19)Af,g2≤∑j∈I0Θjf,g2≤∑j∈IΘjf,g2≤∑j∈JΘjf,g2≤Bf,g2.
Case  2. Let I be any infinite subset of J and f,g∈H⊕KI. Then, for any ε>0, there exists a finite subset I1⊂I and gj∈Hj,j∈I1, such that f,g-∑j∈I1Θj∗gj≤ε. Denote (20)H⊕KI1=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj:∀gj∈Hj,j∈I1¯.Clearly, H⊕KI=H⊕KI1⊕H⊕KI1⊥. Suppose that there exists f1,g1∈H⊕KI1 and f2,g2∈H⊕KI1⊥ such that f,g-∑j∈I1Λj∗gj,∑j∈I1Γj∗gj=f1,g1+f2,g2. By (f,g)-∑j∈I1Θj∗gj≤ε, we get f1,g12+f2,g22≤ε2. Let f0,g0=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj+f1,g1, we get f,g=f0,g0+f2,g2, where f0,g0∈H⊕KI1. By Case  1, we have (21)Af,g2=Af0,g02+Af2,g22≤∑j∈I1Θjf0,g02+Aε2.
Now we prove∑j∈I1Θjf,g2=∑j∈I1Θjf0,g02. We only need to prove that Θjf,g=Θjf0,g0 for every j∈I1. From (6), we obtain (22)Θjf,g=∑k∈Kjf,g,ψjk,φjkejk=∑k∈Kjf0,g0+f2,g2,ψjk,φjkejk,∀j∈I1.By Lemma 10, we have ψjk,φjk∈H⊕KI1 for any j∈I1, k∈Kj. From (22), we get (23)Θjf,g=∑k∈Kjf0,g0,ψjk,φjkejk,∀j∈I1.Again by (6), we have (24)Θjf,g=∑k∈Kjf0,g0,ψjk,φjkejk=Θjf0,g0,∀j∈I1.Using (21), we get (25)Af,g2≤∑j∈I1Θjf0,g02+Aε2=∑j∈I1Θjf,g2+Aε2≤∑j∈IΘjf,g2+Aε2≤∑j∈JΘjf,g2+Aε2≤Bf,g2+Aε2.Letting ε→0, Af,g2≤∑j∈IΘjf,g2≤Bf,g2 for any f,g∈H⊕KI. The proof of Theorem 14 is completed.


### 3.2. Characterization of g-Riesz Frame for Super Hilbert SpaceL1⊕L2⊕⋯⊕Ln

Definition 15.
LetΘj∈LL1⊕L2⊕⋯⊕Ln,Hj:j∈J be a g-frame for L1⊕L2⊕⋯⊕Ln with respect to Hjj∈J. If any subsequence Θjj∈I,I⊂J is also a g-frame for L1⊕L2⊕⋯⊕LnI with respect to Hjj∈I with uniform g-frame lower bound, then one says that Θjj∈J is a g-Riesz frame for L1⊕L2⊕⋯⊕Ln with respect to Hjj∈J.
Like (8), we can define L1⊕L2⊕⋯⊕LnI as follows: (26)L1⊕L2⊕⋯⊕LnI=∑j∈I1Θj∗gj=∑j∈I1Λj1∗gj,∑j∈I1Λj2∗gj,…,∑j∈I1Λjn∗gj:I1⊂I,∀gj∈Hj,j∈I1¯,where I1 is any finite subset of I and Λjp∈LLp,Hj:j∈J, p=1,2,…,n. Similarly, we give two lemmas.Lemma 16.
For everyΘj∈LL1⊕L2⊕⋯⊕Ln,Hj:j∈J, let ejkk∈Kj be an orthonormal basis of Hj and ψjkpp=Λjp∗ejkp, where Λjp∈LLp,Hj:j∈J, p=1,2,…,n. Then (27)L1⊕L2⊕⋯⊕LnI=span¯⁡ψjk11,ψjk22,…,ψjknnj∈I,k1,k2,…,kn∈Kj.Lemma 17.
LetW be a closed subspace of L1⊕L2⊕⋯⊕Ln and let ejkk∈Kj be an orthonormal basis of Hj. Suppose that I is a subset of J and {Θj∈L(L1⊕L2⊕⋯⊕Ln,Hj),j∈J}. For every j∈I, k1,k2,…,kn∈Kj, one has ψjk1(1),ψjk2(2),…,ψjkn(n)∈W, where ψjkpp=Λjp∗ejkp. If Θjj∈I is g-complete in W, then ψjk11,ψjk22,…,ψjkn(n)j∈I,k1,k2,…,kn∈Kj is complete in W.In terms of the concept of minimal g-complete of g-frame for Hilbert space, we give the definition of minimal g-complete of g-frame for super Hilbert spaceL1⊕L2⊕⋯⊕Ln.Definition 18.
LetW be a closed subspace of L1⊕L2⊕⋯⊕Ln and I⊂J. If Θjj∈I is g-complete in W, but Θjj∈I∖j0 is not g-complete in W for any j0∈I, and then one says that Θjj∈I is minimal g-complete in W.Lemma 19.
Suppose thatΘj∈LL1⊕L2⊕⋯⊕Ln,Hj:j∈J and I is any finite nonempty subset of J. Then there exists a finite nonempty subset I0⊂I such that Θjj∈I0 is minimal g-complete in L1⊕L2⊕⋯⊕LnI0 and (28)L1⊕L2⊕⋯⊕LnI0=L1⊕L2⊕⋯⊕LnI,where L1⊕L2⊕⋯⊕LnI is defined as (26).Proof.
The proof is similar to proof of Lemma10.By above lemma, we can get a characterization of g-Riesz frame for super Hilbert spaceL1⊕L2⊕⋯⊕Ln. Obviously, Theorem 14 is the special case of Theorem 20.Theorem 20.
LetΘj∈LL1⊕L2⊕⋯⊕Ln,Hj:j∈J be a g-frame for L1⊕L2⊕⋯⊕Ln with respect to Hjj∈J. Then the following two statements are equivalent.(1)
Θjj∈J is a g-Riesz frame for L1⊕L2⊕⋯⊕Ln with respect to Hjj∈J.(2)
For any nonempty subsetI of J, there exists A>0, if Θjj∈J is minimal g-complete in L1⊕L2⊕⋯⊕LnI. And (29)Af1,f2,…,fn2≤∑j∈IΘjf1,f2,…,fn2,∀f1,f2,…,fn∈L1⊕L2⊕⋯⊕LnI.Proof.
The proof is analogous to proof of Theorem14.


## 3.1. Characterization of g-Riesz Frame for Super Hilbert SpaceH⊕K

Before giving the characterization of g-Riesz frames for super Hilbert spaceH⊕K, we give the definition of g-Riesz frames and some related lemmas.Suppose thatI is a subset of J, and denote (8)H⊕KI=∑j∈I1Θj∗gj=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj:for any finiteI1⊂I,gj∈Hj,j∈I1¯.Definition 9.
Suppose thatΘj∈LH⊕K,Hj:j∈J is a g-frame for H⊕K with respect to Hjj∈J. One says that Θjj∈J is a g-Riesz frame if every subfamily Θjj∈I of Θjj∈J is a g-frame for H⊕KI with respect to Hjj∈I with uniform g-frame lower bounds.For the aboveH⊕KI, we have the following.Lemma 10.
Suppose that for everyj∈J, Λj∈LH,Hj, Γj∈LK,Hj, Θj∈L(H⊕K,Hj), and ejkk∈Kj is an orthonormal basis for Hj. ψjk=Λj∗ejk and φjm=Γj∗ejm are defined as in (5). Then H⊕KI=span¯ψjk,φjmj∈I,k,m∈Kj.Proof.
Denote(9)H⊕KI^=∑j∈I1Θj∗gj=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj:for any finiteI1⊂I,gj∈Hj,j∈I1.Since ψjk=Λj∗ejk, φjm=Γj∗ejm, j∈I, k,m∈Kj, we have ψjk,φjm∈H⊕KI^, j∈I, k,m∈Kj. This implies that spanψjk,φjmj∈I,k,m∈Kj⊂H⊕KI^. Therefore, span¯ψjk,φjmj∈I,k,m∈Kj⊂H⊕KI.
On the other hand, suppose thatf,g∈H⊕KI, and then there exists a finite subset I1⊂I and gj∈Hj, j∈I1, such that f,g=∑j∈I1Θj∗gj=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj. For every j∈I1, let gj=∑k∈Kj‍cjkejk, where cjkk∈Kj∈l2. Then, we have (10)Θj∗gj=Λj∗∑k∈Kjcjkejk,Γj∗∑k∈Kjcjkejk=∑k∈Kjcjkψjk,∑k∈Kjcjkφjk∈span¯⁡ψjk,φjmj∈I,k,m∈Kj.
Therefore,(11)f,g=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj∈span¯⁡ψjk,φjmj∈I,k,m∈Kj.This implies that H⊕KI⊂span¯ψjk,φjmj∈I,k,m∈Kj. The proof of Lemma 10 is completed.Lemma 11.
LetW be a closed subspace of H⊕K and {Θj∈L(H⊕K,Hj)}j∈J. Suppose that ψjk,φjm∈W for every j∈I, k,m∈Kj, where I is a subset of J and ψjkj∈J,k∈Kj, φjmj∈J,m∈Kj are defined as in (5). If Θjj∈I is g-complete in W, then {(ψjk,φjm)}j∈I,k,m∈Kj is complete in W.Proof.
For anyj∈I, k,m∈Kj, we have ψjk,φjm∈W, where W is a closed subspace of H⊕K. Furthermore, we obtain (12)span¯⁡ψjk,φjmj∈I,k,m∈Kj⊂W.It is enough to prove that if f,g∈W and f,g⊥span¯ψjk,φjmj∈I,k,m∈Kj, then f,g,ψjk,φjm=0 for j∈I, k,m∈Kj. By equality (6), we obtain (13)Θjf,g=Λjf+Γjg=∑k∈Kjf,ψjkejk+∑k∈Kjg,φjkejk=0,∀j∈I.Since Θjj∈I is g-complete in W, we have f,g=0,0. Hence, ψjk,φjmj∈J,k,m∈Kj is complete in W.Inspired by the concept of minimal g-complete of g-frame for Hilbert space, we give the definition of minimal g-complete of g-frame for super Hilbert space.Definition 12.
LetW be a closed subspace of H⊕K and I be a subset of J. If Θjj∈I is g-complete in W, but Θjj∈I∖j0 is not g-complete in W for any j0∈I, then one says that Θjj∈I is minimal g-complete in W.Lemma 13.
Suppose thatΘj∈LH⊕K,Hj for j∈J and I is any finite nonempty subset of J. Then there exists a finite nonempty subset I0⊂I such that Θjj∈I0 is minimal g-complete in H⊕KI0 and H⊕KI0=H⊕KI, where H⊕KI is defined as in (8).Proof.
We prove Lemma13 in two cases.
Case  1. Θjj∈I is g-complete in H⊕KI for any j0∈I, but Θjj∈I∖j0 is not g-complete in H⊕KI. Let I0=I. Then the conclusion is right.
Case  2. Suppose that there exists j0∈I such that Θjj∈I∖j0 is g-complete in H⊕KI. Let I1=I∖j0. If there still exists j1∈I1 such that ψjk,φjmj∈I1∖j1,k,m∈Kj is complete in span¯ψjk,φjmj∈I,k,m∈Kj, then we remove j2 from I1 in the same way. Repeat the operation above. Because I is nonempty and finite, this process must stop after finite steps. Assume that we remove j0,j1,…,jn from I, where n∈N and I0=I∖j0,j1,…,jn. Then, we obtain that I0⊂I satisfies the following two statements:
(
1
)
Θ
j
j
∈
I
0 is g-complete in H⊕KI.
(
2
) For any i0∈I0, Θjj∈I0∖i0 is not g-complete in H⊕KI.
By Lemmas10 and 11, the statement (1) implies that
(
3
)
ψ
j
k
,
φ
j
m
j
∈
I
0
,
k
,
m
∈
K
j is complete in span¯ψjk,φjmj∈I,k,m∈Kj.
Obviously,I0 is nonempty. The proof is by contradiction.
SupposeI0=∅, by (1), we obtain that Θjj∈I0 is g-complete in H⊕KI. It is obvious that this is impossible. So I0 is nonempty. Now we prove H⊕KI0=H⊕KI. By Definition 12 and Lemma 10, Θjj∈I is minimal g-complete in span¯ψjk,φjmj∈I,k,m∈Kj. By (3), we get (14)span¯⁡ψjk,φjmj∈I0,k,m∈Kj=span¯⁡ψjk,φjmj∈I,k,m∈Kj.By Lemma 10, we get H⊕KI0=H⊕KI. The proof of Lemma 13 is completed.Based on this, we can obtain a characterization of g-Riesz frame for super Hilbert spaceH⊕K.Theorem 14.
LetΘj∈LH⊕K,Hj:j∈J be a g-frame for H⊕K with respect to Hjj∈J. Then the following two statements are equivalent.
(1)Θjj∈J is a g-Riesz frame for H⊕K with respect to Hjj∈J.
(2) There existsA>0 such that Θjj∈I is minimal g-complete in H⊕KI for any nonempty subset I of J. And (15)Af,g2≤∑j∈IΘjf,g2,∀f,g∈H⊕KI,where H⊕KI is defined as in (8).Proof.
(
1
)
⇒
(
2
). Since Θjj∈J is a g-Riesz frame for H⊕K with respect to Hjj∈J, there exists A>0 such that Θjj∈I is a g-frame for H⊕KI with respect to Hjj∈I for any nonempty I of J. Then, we obtain (16)Af,g2≤∑j∈IΘjf,g2,∀f,g∈H⊕KI.
(
2
)
⇒
(
1
). Suppose that Θjj∈J is a g-frame for H⊕K with respect to Hjj∈J with upper bound B. Then (17)∑j∈JΘjf,g2≤Bf,g2,∀f,g∈H⊕K.
There are two cases to prove thatΘjj∈I is a g-frame for H⊕KI with respect to Hjj∈I, where I is any nonempty subset of J.
Case  1. When I is any finite nonempty subset of J, by Lemma 13, there exists a finite nonempty subset I0⊂I such that Θjj∈I0 is minimal g-complete in H⊕KI0 and H⊕KI0=H⊕KI. By (15), we have (18)Af,g2≤∑j∈I0Θjf,g2,∀f,g∈H⊕KI0=H⊕KI.
Again by (15), for ∀f,g∈H⊕KI, we have (19)Af,g2≤∑j∈I0Θjf,g2≤∑j∈IΘjf,g2≤∑j∈JΘjf,g2≤Bf,g2.
Case  2. Let I be any infinite subset of J and f,g∈H⊕KI. Then, for any ε>0, there exists a finite subset I1⊂I and gj∈Hj,j∈I1, such that f,g-∑j∈I1Θj∗gj≤ε. Denote (20)H⊕KI1=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj:∀gj∈Hj,j∈I1¯.Clearly, H⊕KI=H⊕KI1⊕H⊕KI1⊥. Suppose that there exists f1,g1∈H⊕KI1 and f2,g2∈H⊕KI1⊥ such that f,g-∑j∈I1Λj∗gj,∑j∈I1Γj∗gj=f1,g1+f2,g2. By (f,g)-∑j∈I1Θj∗gj≤ε, we get f1,g12+f2,g22≤ε2. Let f0,g0=∑j∈I1Λj∗gj,∑j∈I1Γj∗gj+f1,g1, we get f,g=f0,g0+f2,g2, where f0,g0∈H⊕KI1. By Case  1, we have (21)Af,g2=Af0,g02+Af2,g22≤∑j∈I1Θjf0,g02+Aε2.
Now we prove∑j∈I1Θjf,g2=∑j∈I1Θjf0,g02. We only need to prove that Θjf,g=Θjf0,g0 for every j∈I1. From (6), we obtain (22)Θjf,g=∑k∈Kjf,g,ψjk,φjkejk=∑k∈Kjf0,g0+f2,g2,ψjk,φjkejk,∀j∈I1.By Lemma 10, we have ψjk,φjk∈H⊕KI1 for any j∈I1, k∈Kj. From (22), we get (23)Θjf,g=∑k∈Kjf0,g0,ψjk,φjkejk,∀j∈I1.Again by (6), we have (24)Θjf,g=∑k∈Kjf0,g0,ψjk,φjkejk=Θjf0,g0,∀j∈I1.Using (21), we get (25)Af,g2≤∑j∈I1Θjf0,g02+Aε2=∑j∈I1Θjf,g2+Aε2≤∑j∈IΘjf,g2+Aε2≤∑j∈JΘjf,g2+Aε2≤Bf,g2+Aε2.Letting ε→0, Af,g2≤∑j∈IΘjf,g2≤Bf,g2 for any f,g∈H⊕KI. The proof of Theorem 14 is completed.


## 3.2. Characterization of g-Riesz Frame for Super Hilbert SpaceL1⊕L2⊕⋯⊕Ln

Definition 15.
LetΘj∈LL1⊕L2⊕⋯⊕Ln,Hj:j∈J be a g-frame for L1⊕L2⊕⋯⊕Ln with respect to Hjj∈J. If any subsequence Θjj∈I,I⊂J is also a g-frame for L1⊕L2⊕⋯⊕LnI with respect to Hjj∈I with uniform g-frame lower bound, then one says that Θjj∈J is a g-Riesz frame for L1⊕L2⊕⋯⊕Ln with respect to Hjj∈J.
Like (8), we can define L1⊕L2⊕⋯⊕LnI as follows: (26)L1⊕L2⊕⋯⊕LnI=∑j∈I1Θj∗gj=∑j∈I1Λj1∗gj,∑j∈I1Λj2∗gj,…,∑j∈I1Λjn∗gj:I1⊂I,∀gj∈Hj,j∈I1¯,where I1 is any finite subset of I and Λjp∈LLp,Hj:j∈J, p=1,2,…,n. Similarly, we give two lemmas.Lemma 16.
For everyΘj∈LL1⊕L2⊕⋯⊕Ln,Hj:j∈J, let ejkk∈Kj be an orthonormal basis of Hj and ψjkpp=Λjp∗ejkp, where Λjp∈LLp,Hj:j∈J, p=1,2,…,n. Then (27)L1⊕L2⊕⋯⊕LnI=span¯⁡ψjk11,ψjk22,…,ψjknnj∈I,k1,k2,…,kn∈Kj.Lemma 17.
LetW be a closed subspace of L1⊕L2⊕⋯⊕Ln and let ejkk∈Kj be an orthonormal basis of Hj. Suppose that I is a subset of J and {Θj∈L(L1⊕L2⊕⋯⊕Ln,Hj),j∈J}. For every j∈I, k1,k2,…,kn∈Kj, one has ψjk1(1),ψjk2(2),…,ψjkn(n)∈W, where ψjkpp=Λjp∗ejkp. If Θjj∈I is g-complete in W, then ψjk11,ψjk22,…,ψjkn(n)j∈I,k1,k2,…,kn∈Kj is complete in W.In terms of the concept of minimal g-complete of g-frame for Hilbert space, we give the definition of minimal g-complete of g-frame for super Hilbert spaceL1⊕L2⊕⋯⊕Ln.Definition 18.
LetW be a closed subspace of L1⊕L2⊕⋯⊕Ln and I⊂J. If Θjj∈I is g-complete in W, but Θjj∈I∖j0 is not g-complete in W for any j0∈I, and then one says that Θjj∈I is minimal g-complete in W.Lemma 19.
Suppose thatΘj∈LL1⊕L2⊕⋯⊕Ln,Hj:j∈J and I is any finite nonempty subset of J. Then there exists a finite nonempty subset I0⊂I such that Θjj∈I0 is minimal g-complete in L1⊕L2⊕⋯⊕LnI0 and (28)L1⊕L2⊕⋯⊕LnI0=L1⊕L2⊕⋯⊕LnI,where L1⊕L2⊕⋯⊕LnI is defined as (26).Proof.
The proof is similar to proof of Lemma10.By above lemma, we can get a characterization of g-Riesz frame for super Hilbert spaceL1⊕L2⊕⋯⊕Ln. Obviously, Theorem 14 is the special case of Theorem 20.Theorem 20.
LetΘj∈LL1⊕L2⊕⋯⊕Ln,Hj:j∈J be a g-frame for L1⊕L2⊕⋯⊕Ln with respect to Hjj∈J. Then the following two statements are equivalent.(1)
Θjj∈J is a g-Riesz frame for L1⊕L2⊕⋯⊕Ln with respect to Hjj∈J.(2)
For any nonempty subsetI of J, there exists A>0, if Θjj∈J is minimal g-complete in L1⊕L2⊕⋯⊕LnI. And (29)Af1,f2,…,fn2≤∑j∈IΘjf1,f2,…,fn2,∀f1,f2,…,fn∈L1⊕L2⊕⋯⊕LnI.Proof.
The proof is analogous to proof of Theorem14.


## 4. Stability of g-Riesz Frames for Super Hilbert SpaceH⊕K

In this section, we use the characterization of g-Riesz frame for super Hilbert spaceH⊕K in Section 3 to study the stability of g-Riesz frame for super Hilbert space H⊕K.The stability of g-frames is important in practice which is wildly studied by many authors; for example, see [22–24]. The following is a fundamental result in the study of the stability of g-frames.Proposition 21 (see [23, Theorem  3.1]).
Suppose thatΛjj∈J is a g-frame for H with respect to Hjj∈J with bounds A and B. There exists λ,β,μ≥0 such that max{λ+μ/A,β}<1. If Γj∈LH,Hjj∈J satisfies (30)∑j∈JΛj-Γjf21/2≤λ∑j∈JΛjf21/2+β∑j∈JΓjf21/2+μffor f∈H, then Γjj∈J is a g-frame for H with respect to Hjj∈J with bounds (31)1-λA-μ1+β2,1+λB+μ1-β2.Example22 illustrates a g-Riesz frame of super Hilbert space H⊕K has no result of stability like Proposition 21.Example 22.
Suppose thatej,0∪0,ej is an orthonormal basis of H⊕H, where j∈J and J=N. Let Hj=span¯ej,ej+1. Define the bounded linear operator as follows: (32)Θjf,f=2Λjf=2f,ejej,∀f,f∈H⊕H.First, we prove thatΘjj∈J is a g-frame for H⊕H with respect to Hjj∈J. In fact, for any f,f∈H⊕H, we have (33)∑j∈JΘjf,f2=∑j∈J2f,ejej2=4∑j∈Jf,ejej2=4f2=2f,f2.Thus, Θjj∈J is a g-frame for H⊕H with respect to Hjj∈J.For anygj∈Hj, let gj=cjej+cj+1ej+1, and we have (34)Θj∗gj,f,f=gj,Θjf,f=gj,2Λjf=cjej+cj+1ej+1,2f,ejej=2cjf,ej¯=2cjej,f=cjej,cjej,f,f.It implies that Θj∗gj=cjej,cjej. Since ej,ej+1 is an orthonormal basis for Hj, we have (35)ψj1=Λj∗ej=ej,ψj2=Λj∗ej+1=0.So ψjk,ψjmj∈J,k,m∈Kj=ej,0,0,ej,0,0,ej,ejj=1∞.Next, we prove thatΘjj∈J is a g-Riesz frame for H⊕H with respect to Hjj∈J. Let I be any finite subset of J. From Lemma 10, we obtain that (36)H⊕HI=span¯⁡ψjk,ψjmj∈I,k,m∈Kj=span¯⁡ej,0,0,ej,0,0,ej,ejj∈I.Hence, for every j0∈J, we have span¯ej,0,0,ejj∈I∖j0≠span¯ej,0,0,ejj∈I. Therefore, ψjk,ψjmj∈I,k,m∈Kj is minimal g-complete in H⊕HI. Now we prove that Θjj∈I is minimal g-complete in H⊕HI.If there existsf,f∈H⊕HI such that Θjf,f=2Λjf=0, then we have (37)0=Θjf,f=2Λjf=2f,ejej=2f,f,ej,0ej,0=Θjf,f=2Λjf=2ej,f¯ej=20,f+ej,f¯ej=20,ej,f,f¯ej.Since ej,ej+1 is an orthonormal basis for Hj, we have f,f,ej,0=0 and 〈(0,ej),(f,f)〉¯=0. From the minimal complete of ψjk,ψjmj∈I for k,m∈Kj in H⊕HI and span¯ψjk,ψjmj∈I,k,m∈Kj=span¯ej,0,0,ejj∈I, we can obtain f,f=0,0. This implies that Θjj∈I is minimal g-complete in H⊕HI. Then, for any f,f∈H⊕HI, we have (38)∑j∈IΘjf,f2=∑j∈I2f,ejej2=2f,f2.By Theorem 14, Θjj∈J is a g-Riesz frame for H⊕H with respect to Hjj∈J.Letε∈0,1. Define the bounded linear operator Ωj:H⊕H→Hj as follows: (39)Ωjf,f=Θjf,f+2εj+1f,ej+1ej+1.By direct calculation, for any gj∈Hj, suppose that gj=cjej+cj+1ej+1, and we have Ωj∗gj=cjej+(ε/(j+1))cj+1ej+1,cjej+(ε/(j+1))cj+1ej+1. Let ψ~j1=ej, ψ~j2=(ε/(j+1))ej+1. Then (40)ψ~jk,ψ~jmj∈J,k,m∈Kj=ej,εj+1ej+1,ej,ej,εj+1ej+1,εj+1ej+1,εj+1ej+1,ejj=1∞.Now we prove thatΩjj∈J is not a g-Riesz frame for H⊕H with respect to Hjj∈J.The proof is by contradiction. Suppose thatΩjj∈J is a g-Riesz frame for H⊕H with respect to Hjj∈J. Let I=j. By Lemma 10, we have (41)H⊕HI=span¯⁡ψjk,ψjmj∈I,k,m∈Kj=span¯⁡ej,εj+1ej+1,ej,ej,εj+1ej+1,εj+1ej+1,εj+1ej+1,ej=span¯⁡ej,ej,ej,ej+1,ej+1,ej,ej+1,ej+1.Choosing ej+1,ej+1∈H⊕HI, then (42)Ωjej+1,ej+12=4ε2j+12.Since Ωjj∈I is a g-frame for (H⊕H)I with respect to Hjj∈I, there exists A>0 such that (43)A=Aej+1,ej+12≤∑k∈IΩkej+1,ej+12=Ωjej+1,ej+12=4ε2j+12.Letting j→+∞, it implies that A=0. But this is a contradiction. We conclude that Ωjj∈J is not a g-Riesz frame for H⊕H with respect to Hjj∈J.On the other hand, for anyf,f∈H⊕H, we have (44)∑j∈JΘj-Ωjf,f21/2=∑j∈J2εj+1f,ej+1ej+121/2≤2εf,f.From Example22, we can realize that λ,β=0, μ≠0. Suppose that Θjj∈J is a g-Riesz frame for H⊕K with respect to Hjj∈J. Even if Θjj∈J and Ωjj∈J satisfy the inequality of Proposition 21, it is uncertain to get that Ωjj∈J is a g-Riesz frame for H⊕K with respect to Hjj∈J.Theorem 23.
LetΘjj∈J be a g-Riesz frame for H⊕K with respect to Hjj∈J with bounds B and A, λ,β∈0,1. If Ωj∈LH⊕K,Hjj∈J satisfies (45)∑j∈J1Θj-Ωjf,g21/2≤λ∑j∈J1Θjf,g21/2+β∑j∈J1Ωjf,g21/2for any f,g∈H⊕K, where J1 is any finite subset of J, then Ωjj∈J is a g-Riesz frame for H⊕K with respect to Hjj∈J with bounds (46)A1-λ1+β2,B1+λ1-β2.Proof.
SinceΘjj∈J is a g-frame for H⊕K with respect to Hjj∈J with bounds A and B, for any finite subset J1⊂J, then (47)∑j∈J1Θjf,g2≤∑j∈JΘjf,g2≤Bf,g2,∀f,g∈H⊕K.By the triangle inequality, we have (48)∑j∈J1Θj-Ωjf,g21/2≥∑j∈J1Ωjf,g21/2-∑j∈J1Θjf,g21/2.From (45), (47), and (48), we have (49)∑j∈J1Ωjf,g21/2≤∑j∈J1Θj-Ωjf,g21/2+∑j∈J1Θjf,g21/2≤λ+11-β∑j∈J1Θjf,g21/2≤λ+1B1-βf,g.Therefore, the series ∑j∈J1Ωjf,g2 is convergent. By (45), we have (50)∑j∈JΘj-Ωjf,g21/2≤λ∑j∈JΘjf,g21/2+β∑j∈JΩjf,g21/2.By Proposition 21, we get that Ωjj∈J is a g-frame for H⊕K with respect to Hjj∈J. For any finite subset J1⊂J, by the triangle inequality, we obtain (51)∑j∈J1Θj-Ωjf,g21/2≥∑j∈J1Θjf,g21/2-∑j∈J1Ωjf,g21/2.Using (45) and (51), we have (52)∑j∈J1Θjf,g21/2≤1+β1-λ∑j∈J1Ωjf,g21/2.
For any finite subsetI⊂J, denote (53)UI=∑j∈IΩj∗gj:∀gj∈Hj,j∈I¯.
IfΩjj∈I is minimal g-complete in UI, we need to prove that UI⊂H⊕KI; that is, H⊕KI⊥⊂UI⊥. For any f,g∈H⊕KI⊥, since H⊕K=UI⊕UI⊥, there exist f1,g1∈UI and f2,g2∈UI⊥ such that f,g=f1,g1+f2,g2. From Lemma 10, we obtain that H⊕KI=span¯ψjk,φjmj∈I,k,m∈Kj and UI=span¯ψ~jk,φ~jmj∈I,k,m∈Kj, where ψ~jk=Λj∗ejk, φ~jm=Γj∗ejm for j∈I, k,m∈Kj. Therefore, for any finite j∈I, we have (54)Θjf,g=∑k∈Kjf,g,ψjk,φjkejk=0,Ωjf,g=∑k∈Kjf,g,ψ~jk,φ~jkejk=∑k∈Kjf1,g1+f2,g2,ψ~jk,φ~jkejk=∑k∈Kjf1,g1,ψ~jk,φ~jkejk=Ωjf1,g1.By (45), we have ∑j∈IΩjf,g21/2≤β∑j∈IΩjf1,g121/2.
Then for anyj∈I, it follows that Ωjf1,g1=0. Since Ωjj∈I is g-complete in UI, we have f1,g1=0. Therefore f,g=f2,g2∈UI⊥, and it implies that H⊕KI⊥⊂UI⊥.
For anyf∈UI⊂H⊕KI, since Θjj∈I is a g-frame for H⊕KI with respect to Hjj∈I with lower bound A, by (52), we have (55)A1-λ1+β2f,g2≤∑j∈IΩjf,g2,∀f,g∈UI.By Theorem 14, we get that Ωjj∈J is a g-Riesz frame for H⊕K with respect to Hjj∈J. The proof of Theorem 23 is completed.



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*Source: 465094-2015-09-30.xml*