# Operator Characterizations and Some Properties ofg-Frames on Hilbert Spaces

Authors: Xunxiang Guo
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Given theg-orthonormal basis for Hilbert space H, we characterize the g-frames, normalized tight g-frames, and g-Riesz bases in terms of the g-preframe operators. Then we consider the transformations of g-frames, normalized tight g-frames, and g-Riesz bases, which are induced by operators and characterize them in terms of the operators. Finally, we discuss the sums and g-dual frames of g-frames by applying the results of characterizations.


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## Body

## 1. Introduction

A sequence(fi)i∈I of elements of a Hilbert space H is called a frame for H if there are constants A,B>0 so that
(1)A∥f∥2≤∑i∈I‍|〈f,fi〉|2≤B∥f∥2,∀f∈H.
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.The concept of frame first appeared in the late 40s and early 50s (see, [1–3]). The development and study of wavelet theory during the last decades also brought new ideas and attentions to frames because of their close connections. There are many related references on this topic, see; [4–8].In [9], 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 K, if there exist two positive constants A and B such that for any f∈H(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 and normalized tight g-frame 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-Bessel sequence, if only the right inequality is satisfied. Recently, g-frames in Hilbert spaces have been studied intensively; for more details see [10–14] and the references therein.Constructing frames and g-frames is an interesting problem in frame theory and it is also useful in applications. In this respect, many mathematicians considered the algebraic operations among frames, which allows us to construct a large number of new frames from existing frames. For more details, see [15–18] and the references therein. In this paper, we will consider the algebraic operations among g-frames; in particular, we will consider the direct sums of g-frames and general sums of g-frames. We also will consider the transforms of g-frames and their dual g-frames.


## 2.g-Preframe Operators and Transformations of g-Frames

In this section, we introduce theg-preframe operators and use them to characterize g-frames. We also discuss the transformations of g-frames. Since any countable infinite sequence can be made as one indexed by natural number N, so all infinite sequences in this paper are assumed to be indexed by N. For each sequence {Hi:i∈N}, we define the space (∑i∈N‍⊕Hi)l2 by
(3)(∑i∈N‍⊕Hi)l2={{fi}i∈N:fi∈Hi,i∈N,∑i∈N‍∥fi∥2<+∞},
with the inner product defined by
(4)〈{fi},{gi}〉=∑i∈N‍〈fi,gi〉.Definition 1 (see, [9]).
LetΛi∈B(H,Hi), i∈N.(i)
If{f:Λif=0;i∈N}={0}, then we say that {Λi:i∈N} is g-complete.(ii)
If{Λi:i∈N} is g-complete and there are positive constants A and B such that for any finite subset J⊂N and gj∈Hj, j∈J,
(5)A∑j∈J‍∥gj∥2≤∥∑j∈J‍Λj*gj∥2≤B∑j∈J‍∥gj∥2,
then we say that {Λi:i∈N} is a g-Riesz basis for H with respect to {Hi:i∈N}.(iii)
We say{Λi:i∈N} is a g-orthonormal basis for H with respect to {Hi:i∈N} if it satisfies the following:
(6)〈Λi*gi,Λj*gj〉=δij〈gi,gj〉,∀i,j∈N,gi∈Hi,gj∈Hj,∑j∈N‍∥Λjf∥2=∥f∥2,∀f∈H.Remark 2.
It is obvious that anyg-frame {Λi:i∈N} is g-complete and any g-orthonormal basis is a normalized tight g-frame.Lemma 3 (see [11]).
{
Λ
i
∈
B
(
H
,
H
i
), i∈N} is g-complete if and only if  span¯{Λi*(Hi)}i∈N=H.Lemma 4 (see, [11]).
{
Λ
i
∈
B
(
H
,
H
i
), i∈N} is a g-Bessel sequence for H with bound B, if and only if the operator
(7)T:(∑i∈N‍⊕Hi)l2⟶H
defined by T({gi})=∑j∈N‍Λj*(gj) is a well-defined bounded operator with ∥T∥≤B.Lemma 5.
Let{θi∈B(H,Hi):i∈N} be a g-orthonormal basis for H with respect to {Hi:i∈N}; then the sequence {Λi∈B(H,Hi):i∈N} is a g-Bessel sequence with respect to {Hi:i∈N} if and only if there is a unique bounded operator V:H→H such that Λi=θiV*, for all i∈N.Proof.
⇒
: Since {θi∈B(H,Hi):i∈N} is a g-orthonormal basis for H, {θif:i∈N}∈(∑i∈N‍⊕Hi)l2, for any f∈H. If {Λi∈B(H,Hi):i∈N} is a g-Bessel sequence, then the operator
(8)V:H⟶H,Vf=∑i∈N‍Λi*θif
is well-defined and bounded operator by Lemma 5. Also by the definition of g-orthonormal basis, it is easy to see that θiθj*g=δijg. So
(9)Vθj*g=∑i∈N‍Λi*θiθj*g=Λj*θjθj*g=Λj*g
for all g∈Hj, j∈N. Hence Vθj*=Λj*, which implies that θjV*=Λj, j∈N. Suppose V1,V2∈B(H) and θiV1*=θiV2*=Λi for any i∈N. Then for any f∈H, gi∈Hi, we have 〈θiV1*f,gi〉=〈θiV2*f,gi〉, that is, 〈V1*f,θi*gi〉=〈V2*f,θi*gi〉 Since span¯{θi*(Hi)}i∈N=H by Lemma 3, V1*f=V2*f, which means that V1=V2. Thus the operator V is unique.
⇐
: Since Λi=θiV*, for all i∈N, for any f∈H, we have
(10)∑i∈N‍∥Λif∥2=∑i∈N‍∥θiV*f∥2=∥V*f∥2≤∥V*∥2∥f∥2.
So {Λi∈B(H,Hi):i∈N} is a g-Bessel sequence with respect to {Hi:i∈N}.Definition 6.
Let{θn:n∈N} be a g-orthonormal basis for H, {Λi∈B(H,Hi):i∈N} is a g-Bessel sequence for H with respect to {Hi:i∈N}, and the  g-preframe operator associated with {Λi:i∈N} is the operator V∈B(H) such that θnV*=Λn for any n∈N. Theg-frame operatorS associated with {Λi∈B(H,Hi):i∈N} is defined by Sf=∑i∈N‍Λi*Λif.Remark 7.
It is well known that if{Λi∈B(H,Hi):i∈N} is a g-frame, then S is a well-defined positive invertible operator on H and {Λ~i=ΛiS-1:i∈N} is the canonical dual g-frame of {Λi:i∈N}; for details see [9]. In this case, for any f∈H, we have the Fourier series type decomposition: f=∑i∈N‍Λi*Λ~if=∑i∈N‍Λ~i*Λif. In general, there are many g-frames {Γi:i∈N} for which the formula f=∑i∈N‍Λi*Γif=∑i∈N‍Γi*Λi*f holds. All such g-frames are called alternate dual g-frames of {Λi:i∈N}.Lemma 8.
Suppose{θi∈B(H,Hi):i∈N} is a g-orthonormal basis for H, {Λi∈B(H,Hi):i∈N} is a g-Bessel sequence for H, and V and S are the g-preframe operator and g-frame operator associated with {Λi∈B(H,Hi):i∈N}, respectively. Then S=VV*.Proof.
SinceΛi=θiV* for all i∈N, for any f∈H we have
(11)Sf=∑i∈N‍Λi*Λif=∑i∈N‍Vθi*θiV*f=V∑i∈N‍θi*θiV*f=VV*f.
So S=VV*.Lemma 9 (see, [11]).
A sequence{θi∈B(H,Hi):i∈N} is a g-frame for H if and only if
(12)S:f⟶∑i∈N‍Λi*Λif
is a well-defined and bounded mapping from H onto H.Remark 10.
SinceS is adjoint, the above condition can be replaced with S is a well-defined and bounded invertible mapping.Lemma 11 (see, [19]).
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
(13)TT†f=f,∀f∈H.Theorem 12.
Suppose{θi∈B(H,Hi):i∈N} is a g-orthonormal basis for H, {Λi∈B(H,Hi):i∈N} is a g-Bessel sequence for H, and V and S are the g-preframe operator and g-frame operator associated with {Λi∈B(H,Hi):i∈N}, respectively. Then(1)
{Λi∈B(H,Hi):i∈N} is a g-frame if and only if V is onto.(2)
{Λi∈B(H,Hi):i∈N} is a normalized tight g-frame if and only if V is a coisometry.(3)
{Λi∈B(H,Hi):i∈N} is a g-Riesz basis if and only if V is invertible.(4)
{Λi∈B(H,Hi):i∈N} is a g-orthonormal basis if and only if V is unitary.Proof.
(1)⇒: By Lemma 9, if {Λi∈B(H,Hi):i∈N} is a g-frame, then S is invertible, but S=VV* by Lemma 8, so V is onto. ⇐: If V is onto, then {Λi∈B(H,Hi):i∈N} is a g-Bessel sequence by Lemma 5. So we just need to show the existence of lower frame bound. Since V is onto, VV†=I by Lemma 11. Hence (V†)*V*=I, so for any f∈H, ∥f∥2=∥(V†)*V*f∥2≤∥(V†)*∥2∥V*f∥2. So we have
(14)∑i∈N‍∥Λif∥2=∑i∈N‍∥θiV*f∥2=∥V*f∥2≥1∥(V†)*∥2∥f∥2.
So {Λi∈B(H,Hi):i∈N} is a g-frame.
(2) Since{Λi∈B(H,Hi):i∈N} is a normalized tight g-frame if and only if S=I, but S=VV* by Lemma 8, so S=I if and only V is coisometry.
(3) If{Λi∈B(H,Hi):i∈N} is a g-Riesz basis then V is invertible by [11, Corollary 2.18]. Conversely, suppose V is invertible. If for any i∈N, Λif=θiV*f=0, since {θi:i∈N} is g-complete, then V*f=0; hence f=0. It follows that {Λi:i∈N} is g-complete. Now for any finite set J⊂N, and gj∈Hj,j∈J, we have
(15)∥∑j∈J‍Λj*gj∥2=∥∑j∈J‍Vθj*gj∥2=∥V∑j∈J‍θj*gj∥2.
Since
(16)∥∑j∈J‍θj*gj∥2=∥VV-1∑j∈J‍θj*gj∥2≤∥V-1∥2∥V∑j∈J‍θj*gj∥2,∥V∑j∈J‍θj*gj∥2≤∥V∥2∥∑j∈J‍θj*gj∥2,∥∑j∈J‍θj*gj∥2=∑j∈J‍∥gj∥2.
So we have
(17)1∥V-1∥2∑j∈J‍∥gj∥2≤∥∑j∈J‍Λj*gj∥2≤∥V∥2∑j∈J‍∥gj∥2.
Hence {Λi∈B(H,Hi):i∈N} is a g-Riesz basis.
(4) If{Λi∈B(H,Hi):i∈N} is a g-orthonormal basis, then V is unitary by [11, Corollary 2.19]. Conversely, suppose that V is unitary. Then for i,j∈N, and gi∈Hi, gj∈Hj we have
(18)〈Λi*gi,Λj*gj〉=〈Vθi*gi,Vθj*gj〉=〈θi*gi,θj*gj〉=δij〈gi,gj〉,∑i∈N‍∥Λi*f∥2=∑i∈N‍∥θiV*f∥2=∥V*f∥2=∥f∥2.
So {Λi∈B(H,Hi):i∈N} is a g-orthonormal basis.Lemma 13.
(1) Suppose thatT,V∈B(H) and V is onto. Then TV is onto if and only if T is onto.
(2) Suppose thatT,V∈B(H) and V is a coisometry. Then TV is a coisometry if and only if T is a coisometry.
(3) Suppose thatT,V∈B(H) and V is invertible. Then TV is invertible if and only if T is invertible.
(4) Suppose thatT,V∈B(H) and V is a unitary. Then TV is unitary if and only if T is unitary.Proof.
(1) IfTV is onto, then for any y∈H, there exists x∈H such that y=TVx=T(Vx), so T is onto. Conversely, if T is onto, then for any y∈H there exists x∈H such that y=Tx. Since V is onto, there exists z∈H such that x=Vz. So y=TVz, which implies that TV is onto.
(2) IfTV is a coisometry, then (TV)(TV)*=I. Since V is a coisometry, VV*=I. So TT*=T(VV*)T*=(TV)(TV)*=I, which implies that T is a coisometry. Conversely, if T is a coisometry, then (TV)(TV)*=TVV*T*=T(VV*)T*=TT*=I. Hence TV is a coisometry.
(3) IfTV is invertible, then T=(TV)V-1, so T is invertible. Conversely, if T is invertible, since V is invertible, then TV is invertible.
(4) IfTV is unitary, then T=TVV*=(TV)V*, so T is unitary. Conversely, if T is unitary, since V is unitary, so TV is unitary.By Theorem12 and Lemma 13, it is easy to get the following results. We leave the details to the readers.Theorem 14.
Suppose that{θi∈B(H,Hi):i∈N} is a g-orthonormal basis for H with respect to {Hi:i∈N}, T∈B(H). Define the transformation ΦT:{Λi∈B(H,Hi):i∈N}→{ΛiT*:i∈N},  then:(1)
It transformsg-frames to g-frames if and only if T is onto.(2)
Ittransforms normalized tightg-frames to normalized tight g-frames if and only if T is a coisometry.(3)
Ittransformsg-Riesz bases to g-Riesz bases if and only if T is invertible.(4)
Ittransformsg-orthonormal bases to g-orthonormal bases if and only if T is unitary.Remark 15.
The above results are established under the constraint thatH has a g-orthonormal basis with respect to {Hi:i∈N}. But in general, the statement (1) in the above Theorem is still true.Theorem 16.
GivenT∈B(H). The transformation ΦT:{Λi∈B(H,Hi):i∈N}→{ΛiT*:i∈N}, transforms g-frames to g-frames if and only if T is onto.Proof.
⇒
: If ΦT transforms g-frames to g-frames, then {ΛiT*:i∈N} is a g-frame whenever {Λi:i∈N} is a g-frame. Suppose that A1,B1 and A2,B2 are the lower and upper frame bounds of {ΛiT*:i∈N} and {Λi:i∈N} respectively. Then for any f∈H, we have
(19)A1∥f∥2≤∑i∈N‍∥ΛiT*f∥2≤B1∥f∥2.
So if T*f=0, then f=0, which implies that T* is one to one. If T*fn→y(n→∞), then {fn} is convergent; that is, there exists y0∈H such that fn→y0. Hence T*fn→T*y0, so y=T*y0, that is, y∈range(T*), which implies that T* has closed range. So T has closed range. Since range(T)=range¯(T)=(kerT*)⊥=H, T is onto.
⇐
: If T is onto and {Λi:i∈N} is a g-frame for H with frame bounds A and B, then for any f∈H, we have
(20)∑i∈N‍∥ΛiT*f∥2≤B∥T*f∥2≤B∥T*∥2∥f∥2
and since TT†=I by Lemma 11,(T†)*T*=I. So
(21)∑i∈N‍∥ΛiT*f∥2≥A∥T*f∥2≥A∥(T†)*∥2∥f∥2.
So {ΛiT*:i∈N} is a g-frame for H, which implies that ΦT transforms g-frames to g-frames.


## 3. Sums ofg-Frames

In this section, we will consider the direct sums and usual sums ofg-frames, which generate g-frames for direct sum of Hilbert spaces and sum of Hilbert spaces, respectively.Definition 17.
SupposeT,L∈B(H,K), then T⊕L∈B(H⊕H,K) is defined by (T⊕L)(x⊕y)=Tx+Ly.Remark 18.
Since for any scalarsa and b, any x,y∈H, we have
(22)(T⊕L)[a(x1⊕y1)+b(x2⊕y2)]=(T⊕L)[(ax1+bx2)⊕(ay1+by2)]=T(ax1+bx2)+L(ay1+by2)=aTx1+bTx2+aLy1+bLy2=a(Tx1+Ly1)+b(Tx2+Ly2)=a(T⊕L)(x1⊕y1)+b(T⊕L)(x2⊕y2).
So T⊕L is linear. Since for any x,y∈H, we have
(23)∥(T⊕L)(x⊕y)∥=∥Tx+Ly∥≤∥Tx∥+∥Ly∥≤∥T∥∥x∥+∥L∥∥y∥≤M(∥x∥+∥y∥)≤M2∥x∥2+∥y∥2=2M∥x⊕y∥,
where M=max{∥T∥,∥L∥}. So T⊕L is bounded. It is easy to see that (T1⊕T2)*z=T1*z⊕T2*z for any z∈Hi. In fact, for any x,y∈H, we have
(24)〈(T1⊕T2)(x⊕y),z〉=〈(x⊕y),(T1⊕T2)*z〉,〈(T1⊕T2)(x⊕y),z〉=〈T1x+T2y,z〉=〈T1x,z〉+〈T2y,z〉=〈x,T1*z〉+〈y,T2*z〉=〈x⊕y,T1*z⊕T2*z〉.
So (T1⊕T2)*z=T1*z⊕T2*z. The finite direct sum of operators have the same meaning.Definition 19.
Let{Λj:j∈N}, {Γj:j∈N} be normalized tight g-frames for Hilbert space H with respect to {Hi:i∈N}, if {Λj⊕Γj:j∈N} is a normalized tight g-frame for H⊕H with respect to {Hi:i∈N}, then we call them  strongly disjoint. More generally, if {Λj:j∈N} and {Γj:j∈N} are g-frames with respect to {Hi:i∈N} such that {Λj⊕Γj:j∈N} is a g-frame for H⊕H with respect to {Hi:i∈N}, then they are called  disjoint.
In general, let{Λ1j:j∈N},…,{Λkj:j∈N} be normalized tight g-frames for Hilbert space H, if {Λ1j⊕⋯⊕Λkj:j∈N} is a normalized g-frame for H⊕H⊕⋯⊕H, which is the inner direct sum of k copies of H, then we call them strongly disjoint. If the k number of sequences are g-frames and their inner direct sum is a g-frame for the inner direct sum of k copies of H, then we call them disjoint.Definition 20.
Let{Λj:j∈N}, {Γj:j∈N} be normalized tight g-frames for Hilbert space H, if {Λj⊕Γj:j∈N} is a g-orthonormal basis of H⊕H, then we call them  strongly complementary.
In general, let{Λ1j:j∈N},…,{Λkj:j∈N} be normalized tight g-frames for Hilbert space H, if {Λ1j⊕⋯⊕Λkj:j∈N} is a g-orthonormal basis for the inner direct sum of k copies of H, then we call them strongly complementary.
In [8], the authors characterize sums of frames by the topological and algebraic properties of P(H), the set of orthogonal projections on H. In this section, we always assume that there exists a g-orthonormal basis {θi:i∈N} for H with respect to {Hi:i∈N} and we will characterize all kinds of sums of g-frames through the properties of g-preframe operators.Theorem 21.
Suppose{Λj:j∈N}, {Γj:j∈N} are normalized tight g-frames for Hilbert space H, T1 and T2 are the g-preframe operators associated with {Λj:j∈N} and {Γj:j∈N}, respectively; then {Λj:j∈N} and {Γj:j∈N} are strongly disjoint if and only if T2T1*=0.Proof.
⇒
: Since {Λj:j∈N} and {Γj:j∈N} are strongly disjoint, so {Λj⊕Γj:j∈N} is a normalized tight g-frame for H⊕H. Since T1, T2 are g-preframe operators associated with {Λj:j∈N} and {Γj:j∈N}, respectively, so T1 and T2 are coisometries by Theorem 12 and Λn=θnT1*, Γn=θnT2* for any n∈N. So for any x,y∈H, we have
(25)∥x∥2+∥y∥2=∥x⊕y∥2=∑i∈N‍∥(Λi⊕Γi)(x⊕y)∥2=∑i∈N‍∥Λix+Γiy∥2=∑i∈N‍∥θiT1*x+θiT2*y∥2=∑i∈N‍∥θi(T1*x+T2*y)∥2=∥T1*x+T2*y∥2=∥x∥2+∥y∥2+2Re〈T2T1*x,y〉.
So Re〈T2T1*x,y〉=0. Replace y by iy in the above argument; then we get Im〈T2T1*x,y〉=0. Hence 〈T2T1*x,y〉=0 for any x,y∈H. Therefore T2T1*=0.
⇐
: Suppose T2T1*=0; then for any x,y∈H, we have
(26)∑i∈N‍∥(Λi⊕Γi)(x⊕y)∥2=∑i∈N‍∥Λix+Γiy∥2=∑i∈N‍∥θiT1*x+θiT2*y∥2=∑i∈N‍∥θi(T1*x+T2*y)∥2=∥T1*x+T2*y∥2=∥T1*x∥2+∥T2*y∥2+2Re〈T1*x,T2*y〉=∥x∥2+∥y∥2+2Re〈T2T1*x,y〉=∥x∥2+∥y∥2=∥x⊕y∥2.
So {Λj⊕Γj:j∈N} is a normalized tight g-frame for H⊕H; that is, {Λn:n∈N} and {Γn:n∈N} are strongly disjoint.It is easy to see that for any finite number of normalized tightg-frames, we have the following result.Corollary 22.
Suppose{Λ1j:j∈N},…,{Λkj:j∈N} are normalized tight g-frames for Hilbert space H, T1,…,Tk are g-preframe operators associated with {Λ1j:j∈N},…,{Λkj:j∈N}, respectively; then {Λ1j:j∈N},…,{Λkj:j∈N} are strongly disjoint if and only if TmTn*=0, m,n=1,…,k.Theorem 23.
Suppose{Λj:j∈N} and {Γj:j∈N} are g-frames for Hilbert space H with respect to {Hi:i∈N}, T1 and T2 are g-preframe operators associated with {Λj:j∈N} and {Γj:j∈N}, respectively. If T2T1*=0, then {Λj:j∈N} and {Γj:j∈N} are disjoint.Proof.
SinceT1 and T2 are g-preframe operators associated with g-frames {Λj:j∈N} and {Γj:j∈N}, respectively, so T1 and T2 are onto operators by Theorem 12. So there exist T1† and T2† such that T1T1†=I and T2T2†=I by Lemma 11. So (T1†)*T1*=I and (T2†)*T2*=I. Hence ∥x∥=∥(T1†)*T1*x∥≤∥(T1†)*∥∥T1*x∥, so ∥T1*x∥≥(1/∥(T1†)*∥)∥x∥. Similarly, ∥T2*x∥≥(1/∥(T2†)*∥)∥x∥. Since T2T1*=0, we have
(27)∑i∈N‍∥(Λi⊕Γi)(x⊕y)∥2=∑i∈N‍∥Λix+Γiy∥2=∑i∈N‍∥θiT1*x+θiT2*y∥2=∑i∈N‍∥θi(T1*x+T2*y)∥2=∥T1*x+T2*y∥2=∥T1*x∥2+∥T2*y∥2+2Re〈T1*x,T2*y〉=∥T1*x∥2+∥T2*y∥2+2Re〈T2T1*x,y〉=∥T1*x∥2+∥T2*y∥2.
Since (28)∥T1*x∥2+∥T2*y∥2≤∥T1*∥2·∥x∥2+∥T2*∥2·∥x∥2≤B·(∥x∥2+∥y∥2)=B·∥x⊕y∥2,
where B=max{∥T1*∥2,∥T2*∥2}. Hence ∑i∈N‍∥(Λi⊕Γi)(x⊕y)∥2≤B·∥x⊕y∥2. Since
(29)∥T1*x∥2+∥T2*y∥2≥1∥(T1†)*∥2∥x∥2+1∥(T2†)*∥2∥y∥2≥A·(∥x∥2+∥y∥2)=A·∥x⊕y∥2,
where A=min{1/∥(T1†)*∥2,1/∥(T2†)*∥2}. Hence
(30)∑i∈N‍∥(Λi⊕Γi)(x⊕y)∥2≥A·∥x⊕y∥2.
So {Λn⊕Γn:n∈N} is a g-frame for H⊕H; that is, {Λj:j∈N} and {Γj:j∈N} are disjoint.Similarly, for any finite number ofg-frames, we have the following.Corollary 24.
Suppose{Λ1j:j∈N},…,{Λkj:j∈N} are g-frames for Hilbert space H, T1,…,Tk are g-preframe operators of {Λ1j:j∈N},…,{Λkj:j∈N}, respectively; If TmTn*=0,m,n=1,…,k, then {Λ1j:j∈N},…,{Λkj:j∈N} are disjoint.Theorem 25.
Let{Λn:n∈N} and {Γn:n∈N} be normalized tight g-frames for H, T1 and T2 are g-preframe operators associated with {Λn:n∈N} and {Γn:n∈N}, respectively. If {Λn:n∈N} and {Γn:n∈N} are strongly complementary, then T2T1*=0 and T1*T1+T2*T2=I.Proof.
If{Λn:n∈N} and {Γn:n∈N} are strongly complementary, then {Λn⊕Γn:n∈N} is a g-orthonormal basis for H⊕H. So they are of course strongly disjoint, therefore T2T1*=0 by Theorem 21. Since {θi:i∈N} is a g-orthonormal basis for H, for any n,k∈N and gn∈Hn, gk∈Hk, we have
(31)〈(Λn⊕Γn)*gn,(Λk⊕Γk)*gk〉=δnk〈gn,gk〉=〈θn*gn,θk*gk〉.
But
(32)〈(Λn⊕Γn)*gn,(Λk⊕Γk)*gk〉=〈Λn*gn⊕Γn*gn,Λk*gk⊕Γk*gk〉=〈T1θn*gn⊕T2θn*gn,T1θk*gk⊕T2θk*gk〉=〈T1θn*gn,T1θk*gk〉+〈T2θn*gn,T2θk*gk〉=〈T1*T1θn*gn,θk*gk〉+〈T2*T2θn*gn,θk*gk〉=〈(T1*T1+T2*T2)θn*gn,θk*gk〉.
Since span¯{θi*gi:i∈N}=H, we have that T1*T1+T2*T2=I.Similarly, for any finite number of normalized tightg-frames, we haveCorollary 26.
Suppose{Λ1j:j∈N},…,{Λkj:j∈N} are normalized tight g-frames for Hilbert space H, T1,…,Tk are g-preframe operators associated with {Λ1j:j∈N},…,{Λkj:j∈N}, respectively, if {Λ1j:j∈N},…,{Λkj:j∈N} are strongly complementary, then TmTn*=0,m,n=1,…,k, and ∑i=1k‍Ti*Ti=I.The above theorems are related with the inner direct sums ofg-frames or normalized tight g-frames. In the following, we will consider the usual sums of g-frames or normalized tight g-frames.Theorem 27.
Suppose that{Λn:n∈N} and {Γn:n∈N} are g-frames for Hilbert space H, T1 and T2 are g-preframe operators associated with {Λn:n∈N} and {Γn:n∈N}, respectively. If T2T1*=0, then {Λn+Γn:n∈N} is a g-frame for H. Moreover, if {Λn:n∈N} and {Γn:n∈N} are normalized tight g-frames and T2T1*=0, then {Λn+Γn:n∈N} is a tight g-frame for H with bounds 2.Proof.
SinceT1 and T2 are g-preframe operators associated with g-frames {Λn:n∈N} and {Γn:n∈N}, respectively, so Λn=θnT1* and Γn=θnT2* for any n∈N, where {θn:n∈N} is a g-orthonormal basis for H. Hence Λn+Γn=θn(T1+T2)* for any n∈N. To show {Λn+Γn:n∈N} is a g-frame, it is sufficient to show that T1+T2 is onto by Theorem 12. Since T2T1*=0, so
(33)(T1+T2)T1*=T1T1*+T2T1*=T1T1*.
Since T1T1* is invertible, so for any z∈H, there exists x=T1*(T1T1*)-1z∈H such that
(34)(T1+T2)x=(T1+T2)T1*(T1T1*)-1z=(T1T1*)(T1T1*)-1z=z.
So T1+T2 is onto.
If{Λn:n∈N} and {Γn:n∈N} are normalized tight g-frames and T2T1*=0, then {Λn+Γn:n∈N} is a g-frame and for any x∈H, we have
(35)∑n∈N‍∥(Λn+Γn)x∥2=∑n∈N‍∥Λnx∥2+∑n∈N∥Γnx∥2‍+∑n∈N‍〈Λnx,Γnx〉+∑n∈N‍〈Γnx,Λnx〉=2∥x∥2+∑n∈N‍〈θnT1*x,θnT2*x〉+∑n∈N‍〈θnT2*x,θnT1*x〉=2∥x∥2+∑n∈N‍〈T2θn*θnT1*x,x〉+∑n∈N‍〈T1θn*θnT2*x,x〉=2∥x∥2+〈T2∑n∈N‍θn*θnT1*x,x〉+〈T1∑n∈N‍θn*θnT2*x,x〉=2∥x∥2+〈T2T1*x,x〉+〈T1T2*x,x〉=2∥x∥2.
So {Λn+Γn:n∈N} is a tight g-frame with bound 2.Similarly, for any finite number ofg-frames, we have the following.Corollary 28.
Suppose{Λ1j:j∈N},…,{Λkj:j∈N} are g-frames for Hilbert space H, T1,⋯,Tk are g-preframe operators associated with {Λ1j:j∈N},…,{Λkj:j∈N}, respectively. If TmTn*=0,m,n=1,…,k, then {Λ1j+⋯+Λkj:j∈N} is a g-frame for H. Moreover, if {Λ1j:j∈N},…,{Λkj:j∈N} are normalized tight g-frames for Hilbert space H and TmTn*=0,m,n=1,…,k, then {Λ1j+⋯+Λkj:j∈N} is a tight g-frame for H with bound k.The above theorem can be generalized further as follows.Theorem 29.
Suppose that{Λn:n∈N} and {Γn:n∈N} are g-frames for Hilbert space H, T1 and T2 are g-preframe operators associated with {Λn:n∈N} and {Γn:n∈N}, respectively, such that T2T1*=0 and A,B∈B(H). If either A or B is onto, then {ΛnA*+ΓnB*:n∈N} is a g-frame for H.Proof.
SinceT1 and T2 are g-preframe operators associated with {Λn:n∈N} and {Γn:n∈N}, respectively, Λn=θnT1* and Γn=θnT2*. So
(36)ΛnA*+ΓnB*=θnT1*A*+θnT2*B*=θn(T1*A*+T2*B*)=θn(AT1+BT2)*.
To show {ΛnA*+ΓnB*:n∈N} is a g-frame it is sufficient to show that AT1+BT2 is onto by Theorem 12. It is well known that T1T1* is invertible. If A is onto, then for any z∈H, there exists x∈H such that Ax=z. So for any z∈H, there exists f=T1*(T1T1*)-1x∈H such that
(37)(AT1+BT2)f=(AT1+BT2)T1*(T1T1*)-1x=AT1T1*(T1T1*)-1x=Ax=z.
It follows that AT1+BT2 is onto. In the case that B is onto, the result can be proved similarly.Similarly, for any finite number ofg-frames, we have the following.Corollary 30.
Suppose{Λ1j:j∈N},…,{Λkj:j∈N} are g-frames for Hilbert space H, and T1,…,Tk are g-preframe operators associated with {Λ1j:j∈N},…,{Λkj:j∈N}, respectively, and TnTm*=0, m,n=1,2,…,k, Ai∈B(H), i=1,…,k. If there exists i∈{1,2,…,k} such that Ai is surjective, then {∑i=1k‍ΛijAi*} is a g-frame for H.


## 4. Dualg-Frames of g-Frames

In this section, we study the dualg-frames by using the properties of g-preframe operators.Theorem 31.
Supposeg-frames {Λn:n∈N} and {Γn:n∈N} are alternate dual g-frame pair for Hilbert space H, T is a coisometry in B(H). Then {ΛnT*} and {ΓnT*:n∈N} are alternate dual g-frame pair for H.Proof.
Since{Λn:n∈N} and {Γn:n∈N} are g-frames for H and T is a coisometry, so {ΛnT*} and {ΓnT*:n∈N} are also g-frames for H by Theorem 14. Since {Λn:n∈N} and {Γn:n∈N} are alternate dual g-frame pair, we have, for any x∈H(38)x=∑n∈N‍Λn*Γnx.
So
(39)Tx=∑n∈N‍TΛn*Γnx=∑n∈N‍(ΛnT*)*Γnx.
Replacing x by T*x and applying TT*=I, we have
(40)x=TT*x=∑n∈N‍(ΛnT*)*ΓnT*x.
So {ΛnT*} and {ΓnT*:n∈N} are alternate dual g-frame pair for H.More general, we have the following result.Theorem 32.
Suppose{Λn:n∈N} and {Γn:n∈N} are alternate dual g-frame pair for Hilbert space H, T is a surjective operator in B(H). Then {ΛnT*:n∈N} and {ΓnT†:n∈N} are alternate dual g-frame pair for H. Similarly, {ΛnT†} and {ΓnT*:n∈N} are alternate dual g-frame pair for H.Proof.
SinceT is onto, TT†=I by Lemma 11, so (T†)*T*=I. It follows that (T†)* is onto. Since {Λn:n∈N} and {Γn:n∈N} are g-frames for H, so {ΛnT*:n∈N}, {ΓnT*:n∈N}, {ΛnT†:n∈N} and {ΓnT†:n∈N} are g-frames for H by Theorem 14. Since {Λn:n∈N} and {Γn:n∈N} are alternate dual g-frame pair, we have, for any x∈H(41)x=∑n∈N‍Λn*Γnx.
So
(42)Tx=∑n∈N‍TΛn*Γnx=∑n∈N‍(ΛnT*)*Γnx.
Replaced x by T†x and applying TT†=I, we have
(43)x=TT†x=∑n∈N‍(ΛnT*)*ΓnT†x.
So {ΛnT*} and {ΓnT†:n∈N} are alternate dual g-frame pair for H. Interchanging Λn and Γn for any n∈N, we get that {ΛnT†} and {ΓnT*:n∈N} are alternate dual g-frame pair for H.Theorem 33.
Let{Λn:n∈N} be a g-frame for H. Let {Λ~n:n∈N} denote the canonical dual g-frame of {Λn:n∈N} and Tx denote the the g-preframe operator associated with {Λn:n∈N}. If P is an orthogonal projection with PTx=TxP, then {ΛnP:n∈N} is a g-frame for PH and the canonical dual g-frame for {ΛnP:n∈N} is {Λ~nP:n∈N}.Proof.
It is easy to see that{ΛnP:n∈N} is a g-frame for PH and {θnP:n∈N} is a g-orthonormal basis for PH with respect to {Hi:n∈N}. We only need to prove the second part. By definition, the canonical dual g-frame for {Λn:n∈N} is {ΛnS-1:n∈N}, where S=TxTx*. So Λ~n=ΛnS-1. Since PTx=TxP, so PTx*=Tx*P. Since Λn=θnTx*, so ΛnP=θnTx*P=θnPTx*. Hence the g-preframe operator associated with {ΛnP:n∈N} is Tx as well. So the canonical dual g-frame for {ΛnP} is {ΛnP(TxTx*)-1={Λn(TxTx*)-1P}={Λ~nP:n∈N}.The following theorem characterizes the alternate dualg-frames in terms of g-preframe operators.Theorem 34.
Let{Λn:n∈N} and {Γn:n∈N} be g-frames for Hilbert space H, and let Tx and Ty be the g-preframe operators associated with {Λn:n∈N} and {Γn:n∈N}, respectively. Then {Λn:n∈N} and {Γn:n∈N} are alternate dual g-frame pair if and only if TyTx*=I or TxTy*=I.Proof.
SinceTx and Ty are g-preframe operators associated with {Λn:n∈N} and {Γn:n∈N}, respectively, so θnTx*=Λn and θnTy*=Γn for any n∈N. Hence for any x∈H,
(44)∑n∈N‍Λn*Γnx=∑n∈N‍Txθn*θnTy*x=TxTy*x.
Similarly, ∑n∈N‍Γn*Λnx=TyTx*x. It follows that {Λn:n∈N and {Γn:n∈N} are alternate dual g-frame pair if and only if TyTx*=I or TxTy*=I.Remark 35.
Throughout this paper we assume that there existsg-orthonormal basis for Hilbert space H. It is well known that any Hilbert space has orthonormal basis. A natural question is that given Hilbert space H, K and sequence of Hilbert space {Hi:i∈J}⊂K, Do the g-orthonormal bases for H with respect to {Hi:i∈J} always exist? if not, what is the condition such that the g-orthonormal bases exist? These interesting questions will be our next research subjects.



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