# g-Bases in Hilbert Spaces

Authors: Xunxiang Guo
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The concept ofg-basis in Hilbert spaces is introduced, which generalizes Schauder basis in Hilbert spaces. Some results about g-bases are proved. In particular, we characterize the g-bases and g-orthonormal bases. And the dual g-bases are also discussed. We also consider the equivalent relations of g-bases and g-orthonormal bases. And the property of g-minimal of g-bases is studied as well. Our results show that, in some cases, g-bases share many useful properties of Schauder bases in Hilbert spaces.


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## Body

## 1. Introduction

In 1946, Gabor [1] introduced a fundamental approach to signal decomposition in terms of elementary signals. In 1952, Duffin and Schaeffer [2] abstracted Gabor’s method to define frames in Hilbert spaces. Frame was reintroduced by Daubechies et al. [3] in 1986. Today, frame theory is a central tool in many areas such as characterizing function spaces and signal analysis. We refer to [4–10] for an introduction to frame theory and its applications. The following are the standard definitions on frames in Hilbert spaces. A sequence {fi}i∈N of elements of a Hilbert space H is called a frame for H if there are constants A,B>0 so that
(1.1)A∥f∥2≤∑i∈N|〈f,fi〉|2≤B∥f∥2.
The numbers A,B are called the lower (resp., upper) frame bounds. The frame is a tight frame if A=B and a normalized tight frame if A=B=1.In [11], Sun raised the concept of g-frame as follows, which generalized the concept of frame extensively. A sequence {Λi∈B(H,Hi):i∈N} is called a g-frame for H with respect to {Hi:i∈N}, which is a sequence of closed subspaces of a Hilbert space V, if there exist two positive constants A and B such that for any f∈H(1.2)A∥f∥2≤∑i∈N∥Λif∥2≤B∥f∥2.
We simply call {Λi:i∈N} a g-frame for H whenever the space sequence {Hi:i∈N} is clear. The tight g-frame, normalized tight g-frame, g-Riesz basis are defined similarly. We call {Λi:i∈N} a g-frame sequence, if it is a g-frame for span¯{Λi*(Hi)}i∈N. We call {Λi:i∈N} a g-Besselsequence, if only the right inequality is satisfied. Recently, g-frames in Hilbert spaces have been studied intensively; for more details see [12–17] and the references therein.It is well known that frames are generalizations of bases in Hilbert spaces. So it is natural to viewg-frames as generalizations of the so-called g-bases in Hilbert spaces, which will be defined in the following section. And that is the main object which will be studied in this paper. In Section 2, we will give the definitions and lemmas. In Section 3, we characterize the g-bases. In Section 4, we discuss the equivalent relations of g-bases and g-orthonormal bases. In Section 5, we study the property of g-minimal of g-bases. Throughout this paper, we use N to denote the set of all natural numbers, Z to denote the set of all integer numbers, and C to denote the field of complex numbers. The sequence of {Hj:j∈N} always means a sequence of closed subspace of some Hilbert space V.


## 2. Definitions and Lemmas

In this section, we introduce the definitions and lemmas which will be needed in this paper.Definition 2.1.
For each Hilbert space sequence{Hi}i∈N, we define the space l2(⊕Hi) by
(2.1)l2(⊕Hi)={{fi}i∈N:fi∈Hi,i∈N,∑i=1+∞∥fi∥2<+∞}.
With the inner product defined by 〈{fi},{gi}〉=∑i=1+∞‍〈fi,gi〉, it is easy to see that l2(⊕Hi) is a Hilbert space.Definition 2.2.
{
Λ
j
∈
B
(
H
,
H
j
)
}
j
=
1
∞ is called g-complete with respect to {Hj} if {f:Λjf=0,forallj}={0}.Definition 2.3.
{
Λ
j
∈
B
(
H
,
H
j
)
}
j
=
1
∞ is called g-linearly independent with respect to {Hj} if ∑j=1∞‍Λj*gj=0, then gj=0, where gj∈Hj(j=1,2,…).Definition 2.4.
{
Λ
j
∈
B
(
H
,
H
j
)
}
j
=
1
∞ is called g-minimal with respect to {Hj} if for any sequence {gj:j∈N} with gj∈Hj and any m∈N with gm≠0, one has Λm*gm∉span¯i≠m{Λi*gi}.Definition 2.5.
{
Λ
j
∈
B
(
H
,
H
j
)
}
j
=
1
∞ and {Γj∈B(H,Hj)}j=1∞ are called  g-biorthonormal with respect to {Hj}, if
(2.2)〈Λj*gj,Γi*gi〉=δj,i〈gj,gi〉,∀j,i∈N,gj∈Hj,gi∈Hi.Definition 2.6.
We say{Λj∈B(H,Hj)}j=1∞ is g-orthonormal basis for H with respect to {Hj}, if it is g-biorthonormal with itself and for any f∈H one has
(2.3)∑j∈N∥Λjf∥2=∥f∥2.Definition 2.7.
We call{Λj∈B(H,Hj)}j=1∞ a g-basis for H with respect to {Hj} if for any x∈H there is a unique sequence {gj} with gj∈Hj such that x=∑j=1∞‍Λj*gj.The following result is about pesudoinverse, which plays an important role in some proofs.Lemma 2.8 (see [5]).
Suppose thatT:K→H is a bounded surjective operator. Then there exists a bounded operator (called the pseudoinverse of T) T†:H→K for which
(2.4)TT†f=f,∀f∈H.The following lemmas characterizeg-frame sequence and g-Bessel sequence in terms of synthesis operators.Lemma 2.9 (see [14]).
A sequence{Λj:j∈N} is a g-frame sequence for H with respect to {Hj:j∈N} if and only if
(2.5)Q:{gj:j∈N}→∑j∈NΛj*gj
is a well-defined bounded linear operator from l2(⊕Hj) into H with closed range.Lemma 2.10 (see [12]).
A sequence{Λj:j∈N} is a g-Bessel sequence for H with respect to {Hj:j∈N} if and only if
(2.6)Q:{gj:j∈N}→∑j∈NΛj*gj
is a well-defined bounded linear operator from l2(⊕Hj) into H.The following is a simple property aboutg-basis, which gives a necessary condition for g-basis in terms of g-complete and g-linearly independent.Lemma 2.11.
If{Λj:j∈N} is a g-basis for H with respect to {Hj:j∈N}, then {Λj:j∈N} is g-complete and g-linearly independent with respect to {Hj:j∈N}.Proof.
SupposeΛif=0 for each i. Then for each gi∈Hi, we have 〈Λif,gi〉=〈f,Λi*gi〉=0. Hence f⊥span{Λi(Hi):i∈N}. Therefore f⊥span¯{Λi(Hi):i∈N}=H. So f=0. So {Λi:i∈N} is g-complete. Now suppose ∑i=1+∞‍Λi*gi=0. Since ∑i=1+∞‍Λi*0=0 and {Λi:i∈N} is a g-basis, so gi=0 for each i. Hence {Λi:i∈N} is g-linearly independent.The following remark tells us thatg-basis is indeed a generalization of Schauder basis of Hilbert space.Remark 2.12.
If{xi}i∈N is a Schauder basis of Hilbert space H, then it induces a g-basis {Λxi:i∈N} of H with respect to the complex number field C, where Λxi is defined by Λxif=〈f,xi〉. In fact, it is easy to see that Λxi*c=c·xi for any c∈C, so for any x∈H, there exists a unique sequence of constants {an:n∈N} such that x=∑i∈N‍aixi=∑i∈N‍Λi*ai.Definition 2.13.
Suppose{Λj:j∈N} is a g-Riesz basis of H with respect to {Hj:j∈N} and {Γj:j∈N} is a g-Riesz basis of Y with respect to {Hj:j∈N}. If there is a homomorphism S:H→Y such that Γj=ΛjS* for each j∈N, then we say that {Λj:j∈N} and {Γj:j∈N} are equivalent.Definition 2.14.
If{Λj} is a g-basis of H with respect to {Hj}, then for any x∈H, there exists a unique sequence {gj:j∈N} such that gj∈Hj and x=∑j=1∞‍Λj*gj. We define a map Γj:H→Hj, by Λjx=gj for each j. Then {Γj} is well defined. We call it the dual sequence of {Λj}, in case that {Γj} is also a g-basis, we call it the dual g-basis of {Λj}.The following results linkg-Riesz basis with g-basis.Lemma 2.15 (see [11]).
A g-Riesz basis{Λj:j∈N} is an exact g-frame. Moreover, it is g-biorthonormal with respect to its dual {Λ~j:j∈N}.Lemma 2.16.
LetΛj∈B(H,Hj),j∈N. Then the following statements are equivalent.(1)
The sequence{Λj}j∈N is a g-Riesz basis for H with respect to {Hj}j∈N.(2)
The sequence{Λj}j∈N is a g-frame for H with respect to {Hj}j∈N and {Λj}j∈N is g-linearly independent.(3)
The sequence{Λj}j∈N is a g-basis and a g-frame with respect to {Hj}j∈N.Proof.
The equivalent between statements(1) and (2) is shown in Theorem2.8 of [12]. By Lemma 2.11, we know that if {Λj}j∈N is a g-basis, then it is g-linearly independent, so (3) implies (2). If {Λj}j∈N is a g-frame for H, then for every x∈H, x=∑j∈N‍Λj*Λ~jx, where {Λ~j}j∈N is the canonical dual g-frame of {Λj}j∈N. Hence for every x∈H, there exists a sequence {gj:j∈N}, gj∈Hj, such that x=∑j∈N‍Λj*gj. Since {Λj}j∈N is g-linearly independent, the sequence is unique. Hence {Λj}j∈N is a g-basis for H. So (2) implies (3).From Lemma2.16, it is easy to get the following well-known result, which is proved more directly.Corollary 2.17.
Suppose{Λj}j∈N is a g-Riesz basis for H, then {Λj}j∈N has a unique dual g-frame.Proof.
It has been shown that everyg-frame has a dual g-frame in [11], so it suffices to show the uniqueness of dual g-frame for g-Riesz bases. Suppose {Γj:j∈N} and {ηj:j∈N} are dual g-frames of {Λj}j∈N. Then for every x∈H, we have x=∑j∈N‍Λj*Γjx=∑j∈N‍Λj*ηjx. Hence ∑j∈N‍Λj*(Γj-ηj)x=0. But {Λj}j∈N is g-linearly independent by Lemma 2.16, so (Γj-ηj)x=0, that is, Γjx=ηjx for each j∈N. Thus, Γj=ηj for each j∈N, which implies that the dual g-frame of {Λj}j∈N is unique.The following lemma generalizes the similar result in frames tog-frames.Lemma 2.18.
Suppose{Λj:j∈N} and {Γj:j∈N} are both g-Bessel sequences for H with respect to {Hj:j∈N}. Then the following statements are equivalent.(1)
For anyx∈H, x=∑j∈N‍Λj*Γjx.(2)
For anyx∈H, x=∑j∈N‍Γj*Λjx.(3)
For anyx,y∈H, 〈x,y〉=∑j∈N‍〈Λjx,Γjx〉.
Moreover, any of the above statements implies that{Λj:j∈N} and {Γj:j∈N} are dual g-frames for each other.Proof.
(
1
)
⇒
(
2
): Since {Λj:j∈N} is a g-Bessel sequence, {Λjx}j∈N∈l2(⊕Hj) for any x∈H. Since {Γj:j∈N} is a g-Bessel sequence, the series ∑j∈N‍Γj*Λjx is convergent by Lemma 2.10. Let x~=∑j∈N‍Γj*Λjx. Then for any y∈H, we have
(2.7)〈x,y〉=〈x,∑j∈NΛj*Γjy〉=∑j∈N‍〈x,Λj*Γjy〉=∑j∈N〈Γj*Λjx,y〉=〈∑j∈NΓj*Λjx,y〉=〈x~,y〉.
So x=x~, that is, (2) is established.
(
2
)
⇒
(
3
): Since for any x∈H, we have x=∑j∈N‍Γj*Λjx; hence for any x,y∈H,
(2.8)〈x,y〉=〈∑j∈NΓj*Λjx,y〉=∑j∈N〈Γj*Λjx,y〉=∑j∈N〈Λjx,Γjx〉.
(
3
)
⇒
(
1
): From the proof of (1)⇒(2), we know that for any x∈H, ∑j∈N‍Γj*Λjx is convergent. Let x~=∑j∈N‍Γj*Λjx, then for any y∈H, we have
(2.9)〈y,x〉=∑j∈N‍〈Λjy,Γjx〉=〈y,∑j∈N‍Γj*Λjx〉=〈y,x~〉.
So x=x~, that is, (1) is true.
If any one of the three statements is true, then for anyx  H, we have
(2.10)∥x∥2=〈∑j∈N‍Λj*Γjx,x〉=∑j∈N‍〈Λjx,Γjx〉≤∑j∈N‍∥Λjx∥∥Γjx∥≤(∑j∈N‍∥Λjx∥2)1/2(∑j∈N‍∥Γjx∥2)1/2≤(B1∥x∥2)1/2(∑j∈N‍∥Γjx∥2)1/2,
where B1 is the bound for the g-Bessel sequence {Λj:j∈N}. So
(2.11)∑j∈N‍∥Γjx∥2≥1B1∥x∥2,
which implies that the g-Bessel sequence {Γj:j∈N} is a g-frame. Similarly, {Λj:j∈N} is also a g-frame. And that they are dual g-frames for each other is obvious by the equality that for any x∈H, x=∑j∈N‍Λj*Γjx.


## 3. Characterizations ofg-Bases

In this section, we characterizedg-bases.Theorem 3.1.
Suppose that{Λj∈B(H,Hj)}j=1∞ is a g-frame sequence with respect to {Hj} and it is g-linearly independent with respect to {Hj}. Let Y={{gj}j=1∞∣gj∈Hj,∑j=1∞‍Λj*gjisconvergent}. If for any {gj}∈Y, set ∥{gj}∥Y=SupN∥∑j=1N‍gj∥, then(1)
Y is a Banach space,(2)
when{Λj} is a g-basis with respect to {Hj} as well, S:Y→H,S({gj})=∑j=1∞‍Λj*gj is a linear bounded and invertible operator, that is, S is a homeomorphism between H and Y.Proof.
(
1
) Let {gj}∈Y, then ∑j=1N‍Λj*gj is convergent as N→∞. Hence {∑j=1N‍Λj*gj}N=1∞ is a convergent sequence, so it is bounded. So ∥{gj}∥Y<∞. It is obvious that for a∈C, {gj}, {hj}∈Y, we have ∥{gj}+{hj}∥Y≤∥{gj}∥Y+∥{hj}∥Y and ∥a·{gj}∥Y=|a|·∥{gj}∥Y. If ∥{gj}∥Y=0, then for any N, ∥∑j=1N‍Λj*gj∥=0, which implies that ∑j=1N‍Λj*gj=0. Since {Λj} is g-linearly independent with respect to {Hj}, we get that gj=0, for j=1,2,…,N. Since N is arbitrary, so {gj}={0}. Thus ∥·∥Y is a norm on Y. Suppose {Gk}k=1∞∈Y is a Cauchy sequence, where Gk={gjk}j=1∞. Then
(3.1)limk,l→∞∥Gk-Gl∥Y=limk,l→∞∥{gjk-gjl}∥Y,(3.2)limk,l→∞SupN∥∑j=1N‍Λj*(gjk-gjl)∥=0.
For any fixed j, we have
(3.3)∥Λj*(gjk-gjl)∥=∥∑t=1j‍Λt*(gtk-gtl)-∑t=1j-1‍Λt*(gtk-gtl)∥≤∥∑t=1j‍Λt*(gtk-gtl)∥+∥∑t=1j-1‍Λt*(gtk-gtl)∥≤2∥Gk-Gl∥Y.
Now let T:l2(⊕Hj)→H,T({gj})=∑j=1∞‍Λj*gj. Since {Λj} is a g-frame sequence with respect to {Hj}, T is a well-defined linear bounded operator with closed range by Lemma 2.9. Since {Λj} is g-linearly independent with respect to {Hj}, T is injective. Hence T*:H→l2(⊕Hj) is surjective. So by Lemma 2.8, there is a bounded operator L, the pseudoinverse of T*, such that T*L=Il2(⊕Hj), which implies that L*T=Il2(⊕Hj). Let {δj} denote the canonical basis of l2(N), then for any gj∈Hj,{gjδj}∈l2(⊕Hj). So {gjδj}=L*T({gjδj})=L*(Λj*gj); hence
(3.4)∥gj∥=∥{gjδj}∥=∥L*(Λj*gj)∥≤∥L∥∥Λj*gj∥.
So by inequalities (3.3), we get
(3.5)∥(gjk-gjl)∥≤∥L∥∥Λj*(gjk-gjl)∥≤2∥L∥∥Gk-Gl∥Y.
So for any fixed j, {gjk}j=1∞ is a Cauchy sequence. Suppose limk→∞gjk=gj. From (3.2), we know that, for any ε>0, there exists L0>0, such that whenever k,l≥L0, we have
(3.6)SupQ∥∑j=1Q‍Λj*(gjk-gjl)∥<ε.
Fix l≥L0, since limk→∞gjk=gj, so whenever l≥L0,
(3.7)SupQ∥∑j=1Q‍Λj*(gj-gjl)∥≤ε.
Since GL0={gjL0}j=1∞∈Y,∑j=1∞‍Λj*gjL0 is convergent. So there exists K0>0, such that whenever M>P≥K0, we have ∥∑j=P+1M‍Λj*gjL0∥<ε. So when M>P>max{L0,K0}, we have
(3.8)∥∑j=P+1M‍Λj*gj∥=∥∑j=1M‍Λj*(gj-gjL0)-∑j=1P‍Λj*(gj-gjL0)+∑j=P+1M‍Λj*gjL0∥≤∥∑j=1M‍Λj*(gj-gjL0)∥+∥∑j=1P‍Λj*(gj-gjL0)∥+∥∑j=P+1M‍Λj*gjL0∥≤3ε.
So ∑j=1∞‍Λj*gj is convergent, thus G={gj}∈Y. Let l→∞ in (3.7), we get that ∥Gl-G∥Y→0. Hence Y is a complete normed space, that is, Y is a Banach space.
(
2
) If {Λj} is a g-basis, then it is g-complete and g-linearly independent with respect to {Hj} by the Lemma 2.11, then the operator S:Y→H,S({gj})=∑j=1∞‍Λj*gj not only is well defined but also is one to one and onto. And for any {gj}∈Y, we have
(3.9)∥S({gj})∥=∥∑j=1∞‍Λj*gj∥=limN→∞∥∑j=1N‍Λj*gj∥≤SupQ∥∑j=1Q‍Λj*gj∥=∥{gj}∥Y.
So S is bounded operator. Since Y is a Banach space, by the Open Mapping Theorem, we get that S is a homeomorphism.Theorem 3.2.
Suppose{Λj:j∈N} is a g-basis of H with respect to {Hj:j∈N} and {Γj:j∈N} is its dual sequence. If {Λj:j∈N} is also a g-frame sequence of H with respect to {Hj:j∈N}, then(1)
∀x∈H, let SNx=∑j=1N‍Λj*Γjx, then SupN∥SNx∥<∞,(2)
C=SupN∥SN∥<∞,(3)
|||x|||=SupN∥SNx∥ is a norm on H and ∥·∥≤|||·|||≤C∥·∥.Proof.
(
1
) Let Y and S be as defined in Theorem 3.1. Then for any x∈H, S-1x={Γjx:j∈N}. So
(3.10)SupN∥SNx∥=SupN∥∑j=1N‍Λj*Γjx∥=∥{Γjx}∥Y=∥S-1x∥≤∥S-1∥∥x∥<∞.
(
2
) Since ∥SNx∥≤SupQ∥SQx∥≤∥S-1∥∥x∥, ∥SN∥≤∥S-1∥. Thus C=SupN∥SN∥<∞.
(
3
) It is obvious that |||·||| is a seminorm. It is sufficient to show that ∥·∥≤|||·|||≤C∥·∥. For any x∈H, we have
(3.11)|||x|||=SupN∥SNx∥≤SupN∥SN∥∥x∥=C∥x∥.
On the other hand,
(3.12)∥x∥=∥∑j=1∞‍Λj*Γjx∥=limN→∞∥∑j=1N‍Λj*Γjx∥≤SupQ∥∑j=1Q‍Λj*Γjx∥=SupQ∥SQx∥=|||x|||.Theorem 3.3.
Suppose{Λj∈B(H,Hj):j∈N} is a g-frame with respect to {Hj:j∈N}. Then {Λj:j∈N} is a g basis with respect to {Hj:j∈N} if and only if there exists a constant C such that for any gj∈Hj, any m,n∈N and m≤n, one has
(3.13)∥∑j=1m‍Λj*gj∥≤C·∥∑j=1n‍Λj*gj∥.Proof.
⇒
: Suppose {Λj:j∈N} is a g-basis with respect to {Hj:j∈N}. Then for any x∈H, there exists a unique sequence {gj:j∈N} with gj∈Hj for each j∈N such that x=∑j=1∞‍Λj*gj. Let |||x|||=Supn∥∑j=1n‍Λj*gj∥. Then by Theorem 3.2, |||·||| is a norm on H and it is equivalent to ∥·∥. So there exists a constant C such that, for any x∈H, |||x|||≤C·∥x∥. Hence for any n∈N, any gj∈Hj, j=1,2,…,n, we choose x=∑j=1n‍Λj*gj, then for any m≤n, we have
(3.14)∥∑j=1m‍Λj*gj∥≤C·∥∑j=1n‍Λj*gj∥.
⇐
: Let 𝔸={∑k∈N‍Λk*gk,gk∈Hk,and∑k∈N‍Λk*gkiscovergent}. First, we show that 𝔸=H. Since {Λj∈B(H,Hj):j∈N}⊂B(H,Hj) is a g-frame, 𝔸 is dense in H. It is sufficient to show that 𝔸 is closed. Suppose {yk}⊂𝔸 and limk→∞yk=y. Denote yk=∑j∈N‍Λj*gj(k). Then for any j∈N and any n≤m≤j, we have, for any k,l∈N,
(3.15)∥Λj*gj(k)-Λj*gj(l)∥≤2C·∥∑s=1m‍Λs*(gs(k)-gs(l))∥≤2C2·∥∑s=1n‍Λs*(gs(k)-gs(l))∥≤2C2·(∥∑s=1n‍Λj*gj(k)-yk∥+∥yk-y∥)+2C2·(∥y-yl∥+∥yl-∑j=1n‍Λs*gs(l)∥).
Since limk→∞yk=y, so for any ε>0, there exists M>0, such that whenever k≥M, we have ∥y-yk∥≤ε/2C2. In the above inequality, let n→∞, we get
(3.16)∥Λj*gj(k)-Λj*gj(l)∥≤εforanyj∈Nandk,l≥M,∥∑s=1m‍Λs*(gs(k)-gs(l))∥≤ε2Cforanym∈Nandanyk,l≥M.
Since {Λj:j∈N} is a g-frame sequence, by inequality (3.4), we have that ∥gj(k)-gj(l)∥≤∥L∥∥Λj*(gj(k)-gj(l))∥≤∥L∥ε for any j∈N and any k,l≥M. So {gj(k)}k∈N is convergent for each j∈N. Suppose limk→∞gj(k)=gj. Then
(3.17)∥Λj*(gj(k)-gj)∥≤εforanyj∈Nandk≥M,∥∑s=1m‍Λs*(gs(k)-gs)∥≤ε2Cforanym∈Nandanyk≥M.
Since
(3.18)∥y-∑j=1m‍Λj*gj∥≤∥y-yk∥+∥yk-∑j=1m‍Λj*gj(k)∥+∥∑j=1m‍Λj*gj(k)-∑j=1m‍Λj*gj∥,
so ∑j=1m‍Λj*gj converges to y, which implies that y∈𝔸. Thus 𝔸 is a closed set. Now we will show that {Λj:j∈N} is g-linearly independent. Suppose that ∑j∈N‍Λj*gj=0, where gj∈Hj for each j∈N. Since for any n∈N and any j≤n, we have ∥Λj*gj∥≤C·∥∑s=1n‍Λs*gs∥, hence ∥Λj*gj∥=0 for any j≤n. But from inequality (3.4), we have ∥gj∥≤∥L∥∥Λj*gj∥. So for each j≤n,gj=0. Since n is arbitrary, gj=0 for any j∈N. Thus {Λj:j∈N} is g-linearly independent. So {Λj:j∈N} is a g-basis.


## 4. Equivalent Relations ofg-Bases

In this section, the equivalent relations ofg-bases were discussed.Theorem 4.1.
Suppose that{Λj:j∈N} is a g-basis of H with respect to {Hj:j∈N}, and S:H→Y is a homeomorphism. Then {ΛjS*:j∈N} is a g-basis of Y with respect to {Hj}.Proof.
For anyy∈Y, S-1y∈H. Since {Λj:j∈N} is a g-basis of H, there exists a unique sequence {gj:j∈N} and gj∈Hj for each j∈N such that S-1y=∑j∈N‍Λj*gj. So y=∑j∈N‍SΛj*gj=∑j∈N‍(ΛjS*)*gj. Suppose there is another sequence {hj:j∈N} and hj∈Hj for each j∈N such that y=∑j∈N‍(ΛjS*)*hj, then y=∑j∈N‍SΛj*hj. So S-1y=∑j∈N‍Λj*hj. But the expansion for S-1y is unique, so hj=gj for each j∈N. Hence {ΛjS*} is a g-basis of Y with respect to {Hj}.Theorem 4.2.
Suppose{Λj:j∈N} is a g-basis of Hilbert space H with respect to {Hj:j∈N}, {Γj:j∈N} is a g-basis of Hilbert space Y with respect to {Hj:j∈N}, and {Gj:j∈N}, and {Lj:j∈N} are dual sequences of {Λj:j∈N} and {Γj:j∈N}, respectively. If {Gj:j∈N} or {Lj:j∈N} is a g-basis, then the following statements are equivalent.(1)
{Λj:j∈N} and {Γj:j∈N} are equivalent.(2)
∑j∈N‍Λj*gj is convergent if and only if ∑j∈N‍Γj*gj is convergent, where gj∈Hj for each j∈N.
Moreover, any one of the above statements implies that both{Gj:j∈N} and {Lj:j∈N} are g-bases and they are also equivalent.Proof.
(
1
)
⇒
(
2
): Suppose there is an invertible bounded operator S:H→Y such that Λj=ΓjS* for each j∈N and ∑j∈N‍Λj*gj is convergent. Then ∑j∈N‍SΓj*gj is convergent. So S-1(∑j∈N‍SΓj*gj)=∑j∈N‍Γj*gj is convergent. Conversely, if ∑j∈N‍Γj*gj is convergent, then ∑j∈N‍S-1Λj*gj is convergent. So S(∑j∈N‍S-1Λj*gj)=∑j∈N‍Λj*gj is convergent.
(
2
)
⇒
(
1
): Without loss of generality, suppose {Gj:j∈N} is a g-basis of H. Then for any x∈H, we have x=∑j∈N‍Λj*Gjx, which is convergent in H, so ∑j∈N‍Γj*Gjx is convergent in Y. Define operator S:H→Y by Sx=∑j∈N‍Γj*Gjx. Then S is well defined and linear. If Sx=0, that is, ∑j∈N‍Γj*Gjx=0, then Gjx=0 for each j∈N. So x=∑j∈N‍Λj*Gjx=0. Hence S is injective. For any y∈Y, y=∑j∈N‍Γj*Ljy, which is convergent in Y. Then ∑j∈N‍Λj*Ljy is convergent in H. Suppose x=∑j∈N‍Λj*Ljy, but we know that x=∑j∈N‍Λj*Gjx and {Λj:j∈N} is a g-basis, so Gjx=Ljy for each j∈N. Hence Sx=∑j∈N‍Γj*Gjx=∑j∈N‍Γj*Ljy=y, which implies that S is surjective. Next, we want to verify that S is bounded.
For anyk∈N, let Tk:H→Y be defined by Tkx=∑j=1k‍Γj*Gjx. Then it is obvious that Tk is well defined and linear. Since
(4.1)∥Tkx∥=∥∑j=1k‍Γj*Gjx∥≤∑j=1k‍∥Γj*Gjx∥≤∑j=1k‍∥Γj*∥∥Gj∥∥x∥,
thus Tk is a bounded operator. It is easy to see that for any x∈H, Tkx→Sx(k→∞). So for any ε>0, there exists k0, such that whenever k≥k0, we have that ∥Tkx-Sx∥<ε. Since for any k∈N, we have ∥Tkx∥≤∥Sx∥+∥Tkx-Sx∥, so for any k∈N, we have
(4.2)∥Tkx∥≤Sup{∥Tjx∥,∥Sx∥+ε∣j=1,2,…,k0-1}<∞.
Hence, by the Banach-Steinhaus Theorem, Supk∥Tk∥<∞. Since for any x∈H, Tkx→Sx(k→∞), we have
(4.3)∥Sx∥≤Supk∥Tkx∥≤Supk∥Tk∥∥x∥.
So S is bounded. Hence S is a bounded invertible operator from H onto Y. Since for any x∈H, we have ∑j∈N‍SΛj*Gjx=∑j∈N‍Γj*Gjx, that is, ∑j∈N‍SΛj*Gj=∑j∈N‍Γj*Gj, so ∑j∈N‍Gj*ΛjS*=∑j∈N‍Gj*Γj. Hence for any y∈Y, we have ∑j∈N‍Gj*ΛjS*y=∑j∈N‍Gj*Γjy. Since {Gj:j∈N} is a g-basis, so ΛjS*=Γj for each j∈N, which implies that {Λj:j∈N} and {Gj:j∈N} are equivalent. In the case that any of the two statements is true, then there is an invertible operator S:H→Y such that Γj=ΛjS* for each j∈N. Hence, for any x∈H,
(4.4)x=∑j∈N‍Γj*Ljx=∑j∈N‍SΛj*Ljx=S(∑j∈N‍Λj*Ljx).
So S-1x=∑j∈N‍Λj*Ljx. Thus x=∑j∈N‍Λj*LjSx, but x=∑j∈N‍Λj*Gjx, and {Λj} is a g-basis of H, it follows that LjS=Gj for each j∈N. By Theorem 4.1, we know that {Lj:j∈N} is also a g-basis and {Gj:j∈N} and {Lj:j∈N} are equivalent.Theorem 4.3.
Suppose{Λj:j∈N} is a g-orthonormal basis for H with respect to {Hj:j∈N}. Then {Λj:j∈N} is a g-basis for H with respect to {Hj:j∈N} and it is self-dual.Proof.
By the definition ofg-orthonormal bases, we know that {Λj:j∈N} is a normalized tight g-frame. So for any f∈H, f=∑j∈N‍Λj*Λjf. Since for any i≠j and for any gj∈Hj,hi∈Hi, we have
(4.5)〈ΛiΛj*gj,hi〉=〈Λj*gj,Λi*hi〉=0,
hence ΛiΛj*gj=0 for i≠j. For i=j and for any gi,hi∈Hi, we have
(4.6)〈ΛiΛi*gi,hi〉=〈Λi*gi,Λi*hi〉=〈gi,hi〉,
so for any gi∈Hi, ΛiΛi*gi=gi. Thus if f=∑j∈N‍Λj*gj, then for any i∈N, we have Λif=∑j∈N‍ΛiΛj*gj=ΛiΛi*gi=gi. Thus for any f∈H, there is a unique sequence {gj:j∈N} such that gj∈Hj for any j∈N and f=∑j∈N‍Λj*gj. So {Λj:j∈N} is a g-basis. It is obvious that {Λj:j∈N} is self-dual.Theorem 4.4.
Suppose{Λj:j∈N} is a g-orthonormal basis with respect to {Hj:j∈N}. Then ∑j∈N‍Λj*gj is convergent if and only if {gj:j∈N}∈l2(⊕Hj).Proof.
Since{Λj:j∈N} is a g-orthonormal basis, by Theorem 4.3, {Λj:j∈N} is a g-basis and a g-frame. So {Λj:j∈N} is a g-Riesz basis. Thus there exist constants A,B>0, such that for any integer n, we have
(4.7)A∑j=1n‍∥gj∥2≤∥∑j=1n‍Λj*gj∥2≤B∑j=1n‍∥gj∥2.
So ∑j∈N‍Λj*gj is convergent if and only if {gj:j∈N}∈l2(⊕Hj).From Theorems4.2, 4.3, and 4.4, the following corollary is obvious.Corollary 4.5.
Any two g-orthonormal bases are equivalent.


## 5. The Property ofg-Minimal of g-Bases

In this section, we studied the property ofg-minimal of g-bases.Theorem 5.1.
Suppose{Λj:j∈N} is a g-frame sequence. Then(1)
if{Λj:j∈N} is a g-basis, then {Λj:j∈N} is g-minimal;(2)
if{Λj:j∈N} is g-minimal, then {Λj:j∈N} is g-linearly independent.Proof.
(
1
). Since {Λj} is a g-basis and it is also a g-frame sequence, it is easy to see that {Λj:j∈N} is a g-frame. Hence {Λj:j∈N} is a g-Riesz basis by (3) of Lemma 2.16. Suppose {Λ~j:j∈N} is the unique dual g-frame of {Λj:j∈N}. By Lemma 2.15, we know that {Λj} and {Λ~j:j∈N} are g-biorthonormal, that is, 〈Λj*gj,Λ~j*gi〉=δij〈gj,gi〉, where gj∈Hj,gi∈Hi. For any m∈N and any sequence {gj:j∈N} with gj∈Hj and gm≠0, let Em=span¯i≠m{Λ~i*gi}. Then for any x∈Em, 〈x,Λ~m*gm〉=0, but 〈Λm*gm,Λ~m*gm〉=〈gm,gm〉≠0, so Λm*gm∉Em. Hence {Λj:j∈N} is g-minimal.
(
2
). Suppose {Λj:j∈N} is g-minimal. If ∑j∈N‍Λj*gj=0, where gj∈Hj for each j∈N, then gj=0 for any j∈N. In fact, if there exists m∈N such that gm≠0, then ∥Λm*gm∥≥(1/L)∥gm∥>0 by inequality (3.4), which implies that Λm*gm≠0. Since Λm*gm=-∑j≠m‍Λj*gj, Λm*gm∈span¯j≠m{Λj*gj}, which contradicts with the fact that {Λj:j∈N} is g-minimal.Theorem 5.2.
Given sequence{Λj∈B(H,Hj):j∈N}.(1)
If there exists a sequence{Γj∈B(H,Hj):j∈N}, such that {Λj:j∈N} and {Γj:j∈N} are biorthonormal, then {Λj:j∈N} is g-minimal.(2)
If there exists a unique sequence{Γj∈B(H,Hj):j∈N} such that {Λj:j∈N} and {Γj:j∈N} are biorthonormal, then {Λj:j∈N} is g-minimal and g-complete.Proof.
The proof of(1) is similar to the proof of (1) of Theorem 5.1, we omit the details. Now we prove (2): By (1) we know that {Λj:j∈N} is g-minimal. So it only needs to show that {Λj:j∈N} is g-complete. Suppose that x∈H and 〈x,Λj*gj〉=0 for any j∈N and any gj∈Hj. Since 〈Λj*gj,Γj*gi〉=δij〈gj,gi〉, so 〈Λj*gj,x+Γjgi〉=δij〈gj,gi〉, which implies that {δx+Γj:j∈N} and {Λj:j∈N} are biorthonormal, where δx*∈B(V,Hj) for each j∈N defined by δx*(f)=x for any f∈V. But it is assumed that there exists a unique sequence {Γj∈B(H,Hj):j∈N} such that {Λj:j∈N} and {Γj:j∈N} are biorthonormal, so δx=0, hence x=δx*(f)=0, which implies that span¯{Λj*gj}=H. So {Λj:j∈N} is g-complete.



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*Source: 923729-2012-12-20.xml*