# Decompositions ofg-Frames and Duals and Pseudoduals of g-Frames in Hilbert Spaces

Authors: Xunxiang Guo
Journal: Journal of Function Spaces
 (2015)

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## Abstract

Firstly, we study the representation ofg-frames in terms of linear combinations of simpler ones such as g-orthonormal bases, g-Riesz bases, and normalized tight g-frames. Then, we study the dual and pseudodual of g-frames, which are critical components in reconstructions. In particular, we characterize the dual g-frames in a constructive way; that is, the formulae for dual g-frames are given. We also give some g-frame like representations for pseudodual g-frame pairs. The operator parameterizations of g-frames and decompositions of bounded operators are the key tools to prove our main results.


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## Body

## 1. Introduction

A sequence(fi)i∈I of elements of a Hilbert space H is called aframe for H if there are constants A,B>0 so that(1)Af2≤∑i∈I‍f,fi2≤Bf2,∀f∈H.The numbers A and B are called thelower (resp.,upper) frame bounds. The frame is atight frame if A=B and anormalized 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 attention 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)Af2≤∑i∈N‍Λif2≤Bf2,where A is called thelowerg-frame bound and B is called theupperg-frame bound. The largest lower frame bound and the smallest upper frame bound are called theoptimal lowerg-frame bound and theoptimal upperg-frame bound, respectively. 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. A g-frame {Γj:j∈N} for H is called analternate dualg-frame of {Λj:j∈N}, if for every f∈H, we have(3)f=∑j∈N‍Λj∗Γjf=∑j∈N‍Γj∗Λjf.If {Λj:j∈N} is a g-frame for H, then the operator S∈B(H) such that (4)Sf=∑j∈N‍Λj∗Λjf,∀f∈His called the g-fame operator associated with {Λj:j∈N}. It is well-known that {ΛjS-1:j∈N} is a dual g-frame of {Λj:j∈N}, which is called thecanonical dualg-frame associated with {Λj:j∈N}. In this paper, we use dual of g-frames to denote any of the duals. Recently, g-frames in Hilbert spaces have been studied intensively; for more details, see [10–16] and the references therein.Frames andg-frames have advantages of allowing decomposing and reconstructing elements in Hilbert spaces, in which the dual and pseudodual of frames (g-frames) play important roles. Characterizing dual frames and general frame decompositions is an important problem in pure and applied fields. In [17–22], the authors study the dual frames and frame-like decompositions in Hilbert spaces. In particular, Li derived a general parametric and algebraic formula for all duals of a frame in [17] and introduced the pseudoframe decompositions in [18]. In this paper, motivated by these works on frames, we consider similar problems on g-frames in Hilbert spaces and generalize the corresponding results on frames to g-frames. Another interesting problem in frame theory is representing general g-frames in terms of special and more simpler g-frames such as g-orthonormal bases, g-Riesz bases, and normalized tight g-frames. In [23], the authors study similar questions on frames in Hilbert spaces by applying the techniques of decomposing linear bounded operators. In this paper, we will study the decompositions of g-frames in Hilbert spaces by using similar techniques combing with what we have obtained on the operator parameterizations for g-frames in [24].Throughout this paper, we useN to denote the set of natural numbers and C to denote the complex plane. All Hilbert spaces in this paper are assumed to be separable complex Hilbert spaces. This paper is organized as follows. In Section 2, we give some definitions and lemmas which are needed to understand the following sections. In Section 3, we consider the decomposition of g-frames. In Section 4, the dual and pseudodual of g-frames are considered.


## 2. Preliminary Definitions and Lemmas

In this section, we introduce some basic definitions and lemmas which are necessary for the following sections.Definition 1.
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∈J‍gj2≤∑j∈J‍Λj∗gj2≤B∑j∈J‍gj2,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=δijgi,gj,∀i,j∈N,gi∈Hi,gj∈Hj,∑j∈N‍Λjf2=f2,∀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.Definition 3.
Suppose thatΛj∈B(H,Hj), Γj∈B(H,Hj) for any j∈N. If, for any x,y∈H, we have x,y=∑j∈N‍Λjx,Γjy, then we call {Λj:j∈N} and {Γj:j∈N}a pair of pseudodualg-frames for H. In particular, if {Λj:j∈N} is a g-frame for H, we call {Γj:j∈N} apseudodualg-frame of {Λj:j∈N}.Lemma 4 (see [25]).
LetH be a Hilbert space. Then,(1)
for every invertible operatorU∈B(H), there exists a unique decomposition U=WP, where W is a unitary operator and P is a positive operator.(2)
for every positive operatorP∈B(H) with P≤1, P=1/2(W+W∗), where W=P+iI-P2 is a unitary operator.Lemma 5 (see [26]).
Given Hilbert spaceH and a sequence of closed subspaces {Hj:j∈N} of a Hilbert space K, then there exists a g-orthonormal basis {Λj∈B(H,Hj):j∈N} for H with respect to {Hj:j∈N} if and only if ∑j∈N‍dim⁡Hj=dim⁡H.Lemma 6 (see [24]).
Let{θj∈B(H,Hj):j∈N} be a g-orthonormal basis for H with respect to {Hj:j∈N}. Then, the sequence {Λj:j∈N} is a g-Bessel sequence for H if and only if there is a unique bounded operator V∈B(H) such that Λj=θjV∗ for all j∈N.Remark 7.
Given theg-orthonormal basis {θi∈B(H,Hi):i∈N}, the operator V in Lemma 6 is called the g-preframe operator associated with {Λi∈B(H,Hi):i∈N}.Lemma 8 (see [24]).
Suppose that{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.Lemma 9 (see [23]).
LetT∈B(H) be onto; then T can be written as a linear combination of two unitary operators if and only if T is invertible.


## 3. Decompositions ofg-Frames

In this section, we do some research on the decompositions ofg-frames in Hilbert spaces by using similar techniques in [23] combing with what we have established on the operator parameterizations for g-frames in [24].Theorem 10.
Suppose that{Λj∈B(H,Hj):j∈N} is a g-Bessel sequence for H. Let T be the g-preframe operator associated with {Λj:j∈N}. Then, for any ɛ∈(0,1), there exist three g-orthonormal bases {θjl:j∈N}  (l=1,2,3) such that Λj=(1-ɛ)-1T(θj1+θj2+θj3) for any j∈N.Proof.
Since we have assumed that all Hilbert spaces are separable, theg-orthonormal bases for H with respect to {Hj:j∈N} exist by Lemma 5. Let {θj∈B(H,Hj):j∈N} be a g-orthonormal basis for H with respect to {Hj:j∈N}. Since {Λj∈B(H,Hj):j∈N} is a g-Bessel sequence for H, there exists a bounded operator T∈B(H) such that Λj=θjT∗ for any j∈N by Lemma 6. Define an operator U∈B(H) by U=1/2I+1-ɛ/2·T∗/T, where I is the identity operator on H. Since(7)I-U=12I-1-ɛ2·T∗T≤12I+1-ɛ2·T∗T=12+1-ɛ2=1-ɛ2<1,U is invertible. By Lemma 4, there exist a unitary V and a positive operator P such that U=VP. Since(8)P=V-1U≤U≤12I+1-ɛ2·T∗T=12+1-ɛ2≤1,P=1/2(W∗+W), where W=P+iI-P2 is a unitary operator by Lemma 4. So(9)T∗=2T1-ɛU-12I=2T1-ɛV2W+W∗-I2=T1-ɛVW+VW∗-I.Hence,(10)Λj=θjT∗=θj·T1-ɛVW+VW∗-I=T1-ɛθjVW+θjVW∗-θj.Denote θj1=θjVW, θj2=θjVW∗, and θj3=-θj for any j∈N. Then, it is easy to see that {θjl:j∈N}  (l=1,2,3) are g-orthonormal bases for H, since V and W are unitary operators. So Λj=(1-ɛ)-1T(θj1+θj2+θj3) for any j∈N.Since ag-frame is of course a g-Bessel sequence, the following corollary is obvious.Corollary 11.
Everyg-frame can be represented as a multiple of sum of three g-orthonormal bases.Theorem 12.
Ag-frame {Λj∈B(H,Hj):j∈N} for H can be written as a linear combination of two g-orthonormal bases for H if and only if {Λj:j∈N} is a g-Riesz basis for H.Proof.
⇒
: Suppose that {Γj:j∈N} and {Lj:j∈N} are g-orthonormal bases for H such that Λj=a·Γj+b·Lj for any j∈N. By Lemma 8, there exist surjective operator T and unitary operator U such that Λj=ΓjT∗ and Lj=ΓjU for any j∈N. So, ΓjT∗=a·Γj+b·ΓjU, ∀j∈N. Hence, TΓj∗=a¯·Γj∗+b¯·U∗Γj∗, ∀j∈N. It implies that T=a¯·I+b¯·U∗, since Span⁡¯{Λj∗(Hj):j∈N}=H. So T is invertible by Lemma 9. Hence, {Λj:j∈N} is a g-Riesz basis for H.
⇐
: Since {Λj:j∈N} is a g-Riesz basis for H, there exist a g-orthonormal basis {θj∈B(H,Hj):j∈N} and an invertible operator T∈B(H) such that Λj=θjT∗ for any j∈N by Lemma 8. There exist two unitary operators U1 and U2 in B(H) and constants a,b such that T∗=a·U1+b·U2 by Lemma 9. So Λj=θjT∗=θj(a·U1+b·U2)=a·θjU1+b·θjU2 for any j∈N. Since {θjU1:j∈N} and {θjU2:j∈N} are g-orthonormal bases for H by Lemmas 8 and 9, the result follows.Theorem 13.
Everyg-frame for H is a multiple of two normalized tight g-frames for H.Proof.
Suppose that{Λj∈B(H,Hj):j∈N} is a g-frame for H and {θj∈B(H,Hj):j∈N} is a g-orthonormal basis for H. Then, there exists a surjective operator T∈B(H) such that Λj=θjT∗ for any j∈N by Lemma 8. Let U=T/2T. Then, U=1/2<1 and U is also surjective. Suppose that U=VP is the polar decomposition of U, where V is a coisometry and P is a positive operator in B(H). Since P=V∗U≤U<1, then P=1/2(W+W∗) with W=P+iI-P2 being a unitary operator. So U=VP=1/2V(W+W∗). It follows that T=2TU=T(VW+VW∗). So(11)Λj=θjT∗=θjTW∗V∗+WV∗=TθjW∗V∗+θjWV∗.Since VW and VW∗ are coisometries, {θjW∗V∗:j∈N} and {θjWV∗:j∈N} are normalized tight g-frames for H by Lemma 8. This finishes the proof.Theorem 14.
Everyg-frame for H is a multiple of the sum of a g-orthonormal basis for H and a g-Riesz basis for H.Proof.
Suppose that{Λj∈B(H,Hj):j∈N} is a g-frame for H and {θj∈B(H,Hj):j∈N} is a g-orthonormal basis for H. Let T be the g-preframe operator associated with {Λj:j∈N}; then Λj=θjT∗ for any j∈N. Define operator S by S=3/4I+1/4(1-ɛ)·T∗/T; then(12)I-S=I4-14·1-ɛ·T∗T≤14·I+1-ɛ4<1,S≤34+1-ɛ4<1.So S is invertible. Let S=VP be the polar decomposition of S. Then, V is a unitary operator and P is a positive operator by Lemma 4. Since P=V∗S, P=V∗S≤V∗·S<1. So P=1/2(W+W∗) by Lemma 4, where W=P+iI-P2 is a unitary operator. So S=VP=1/2(VW+VW∗). It implies that(13)T∗=4T1-ɛS-34·I=4T1-ɛ12·VW+VW∗-34·I=2T1-ɛVW+VW∗-32·I=2T1-ɛVW+R,where R=VW∗-3/2·I. Since(14)I--12·R=I4+12VW∗≤I4+12·VW∗=14+12=34<1,-1/2·R is invertible. Hence, R is invertible. So(15)Λj=θjT∗=θj·2T1-ɛ·VW+R=2T1-ɛθjVW+θjR.Since {θjVW:j∈N} is a g-orthonormal basis for H and {θjR:j∈N} is a g-Riesz basis for H by Lemma 8, {Λj:j∈N} is a multiple of a sum of a g-orthonormal basis and a g-Riesz basis for H.


## 4. Dual and Pseudodualg-Frames

In this section, we consider the characterizations of dual and pseudodualg-frames. The algebraic formula about the dual of g-frames for a given g-frame will be given and some properties on dual and pseudodual g-frames will be established.Theorem 15.
Suppose that{Λj∈B(H,Hj):j∈N} is a g-frame for H and {θj∈B(H,Hj):j∈N} is a g-orthonormal basis for H. Suppose that the g-preframe operator associated with {Λj:j∈N} is T; that is, Λj=θjT∗ for any j∈N. Then, {Γj∈B(H,Hj):j∈N} is a dual g-frame of {Λj:j∈N} if and only if Γj=θjV∗ for any j∈N, where V is a bounded left inverse of T∗.Proof.
⇒
: Suppose that {Γj:j∈N} is a dual g-frame of {Λj:j∈N}. Let V be the g-preframe operator of {Γj:j∈N}. Then, Γj=θjV∗ for any j∈N and V is bounded. Since, for any f∈H, we have f=∑j∈N‍Λj∗Γjf, (16)f=∑j∈N‍Tθj∗θjV∗f=T∑j∈N‍θj∗θjV∗f=TV∗f.It implies that TV∗=I. Hence, VT∗=I. It follows that V is a bounded left inverse of T∗.
⇐
: Since VT∗=I, V is bounded surjective operator in B(H). Hence, {Γj:j∈N} is a g-frame for H by Lemma 8. Since(17)∑j∈N‍Λj∗Γjf=∑j∈N‍Tθj∗θjV∗f=T∑j∈N‍θj∗θjV∗f=TV∗f=f,∀f∈H,{Γj:j∈N} is a dual g-frame for H.Lemma 16.
Suppose that{Λj∈B(H,Hj):j∈N} is a g-frame for H, whose g-preframe operator is T. Then, V is a linear bounded left inverse of T∗ if and only if(18)V=S-1T+WI-T∗S-1T,where S is the g-frame operator associated with {Λj:j∈N}, W∈B(H), and  I is the identity operator in B(H).Proof.
⇒
: Suppose V is a linear bounded left inverse of T∗. Let W=V. Then(19)S-1T+VI-T∗S-1T=S-1T+V-VT∗S-1T=S-1T+V-S-1T=V.⇐: Suppose V=S-1T+W(I-T∗S-1T). Then(20)VT∗=S-1TT∗+WI-T∗S-1TT∗=S-1S+WT∗-T∗S-1TT∗=I.Hence, V is a linear bounded left inverse of T∗.Theorem 17.
Suppose{Λj∈B(H,Hj):j∈N} is a g-frame for H, T is its g-preframe operator, and S is its g-frame operator. Let {θj∈B(H,Hj):j∈N} be a g-orthonormal basis for H. Then, {Γj∈B(H,Hj):j∈N} is a dual g-frame of {Λj:j∈N} if and only if there exists a bounded operator W∈B(H) such that(21)Γj=ΛjS-1+θjW∗-ΛjS-1TW∗,∀j∈N.Proof.
⇒
: Suppose that {Γj∈B(H,Hj):j∈N} is a dual g-frame of {Λj∈B(H,Hj):j∈N}. Then, by Theorem 15, we know that Γj=θjV∗ for any j∈N, where V is a linear bounded left inverse of T∗. By Lemma 16, V=S-1T+W(I-T∗S-1T) for some linear bounded operator W∈B(H). Hence, for any j∈N, we have(22)Γj=θjV∗=θjS-1T+WI-T∗S-1T∗=θjT∗S-1+W∗-T∗S-1TW∗=θjT∗S-1+θjW∗-θjT∗S-1TW∗=ΛjS-1+θjW∗-ΛjS-1TW∗.⇐: Suppose that there exists a linear bounded operator W∈B(H) such that Γj=ΛjS-1+θjW∗-ΛjS-1TW∗. Then,(23)Γj=θjT∗S-1+θjW∗-θjT∗S-1TW∗=θjT∗S-1+W∗-T∗S-1TW∗=θjS-1T+W-WT∗S-1T∗.So {Γj:j∈N} is a g-Bessel sequence for H and the g-preframe operator associated with {Γj:j∈N} is(24)V=S-1T+W-WT∗S-1T=S-1T+WI-T∗S-1T.Since V is a linear bounded left inverse of T∗ by Lemma 16, VT∗=I. Therefore, {Γj:j∈N} is a dual g-frame of {Λj:j∈N} by Theorem 15.Theorem 18.
Suppose that{Λj∈B(H,Hj):j∈N} is a g-frame for H. If {Γj∈B(H,Hj):j∈N} is a pseudodual g-frame of {Λj:j∈N}, then {Γj:j∈N} has lower g-frame bound.Proof.
Since{Γj:j∈N} is a pseudodual g-frame of {Λj:j∈N}, x,y=∑j∈N‍Λjx,Γjy for any x,y∈H. In particular, x,x=∑j∈N‍Λjx,Γjx; that is, x2=∑j∈N‍Λjx,Γjx. Since(25)∑j∈N‍Λjx,Γjx≤∑j∈N‍Λjx·Γjx≤∑j∈N‍Λjx21/2·∑j∈N‍Γjx21/2≤Bx21/2·∑j∈N‍Γjx21/2,where B is the upper g-frame bound of {Λj:j∈N}, hence,(26)∑j∈N‍Γjx2≥1B·x2.So {Γj:j∈N} has lower g-frame bound.Corollary 19.
Suppose that{Λj∈B(H,Hj):j∈N} is a g-frame for H and {Γj:j∈N} is a pseudodual g-frame of {Λj:j∈N}; then {Γj:j∈N} is g-complete.Proof.
Since{Γj:j∈N} has lower g-frame bound by Theorem 18, there exists a constant B>0 such that(27)∑j∈N‍Γjx2≥Bx2,∀x∈H.If Γjx=0, for all j∈N, then Bx2=0; it follows that x=0. So {Γj:j∈N} is g-complete.Theorem 20.
Suppose that{Λj∈B(H,Hj):j∈N} and {Γj∈B(H,Hj):j∈N} are a pair of pseudodual g-frames for H. Then, for any x∈H, x=∑j∈N‍Λj∗Γjx if and only if x=∑j∈N‍Γj∗Λjx, where the series converge in norm of H.Proof.
It is obvious that we only need to prove one direction; the other direction is identical. Now, suppose thatx=∑j∈N‍Λj∗Γjx. Since {Λj∈B(H,Hj):j∈N} and {Γj∈B(H,Hj):j∈N} are a pair of pseudodual g-frames for H, we have x,y=∑j∈N‍Λjx,Γjy∀x,y∈H. It is obvious that fN(y)=∑j=1N‍Λjx,Γjy-x,y is a weakly continuous function on H and limN→∞fN(y)=0 for each x∈H. Since the closed unit ball of H is weakly compact, for any ɛ>0, there exists N0>0 such that, for any y≤1 and any N>N0, we have fN(y)<ɛ. So whenever N>N0, we have (28)∑j=1N‍Γj∗Λjx-x=sup⁡y=1⁡∑j=1N‍Γj∗Λjx-x,y=sup⁡y=1⁡∑j=1N‍Γj∗Λjx,y-x,y=sup⁡y=1⁡fNy≤ɛ.Hence, x=∑j∈N‍Γj∗Λjx.Corollary 21.
Suppose that{Λj∈B(H,Hj):j∈N} and {Γj∈B(H,Hj):j∈N}are a pair of pseudo g-frames, x0∈H. If ∑j∈N‍Λj∗Γjx0 is convergent, then(29)x0=∑j∈N‍Λj∗Γjx0=∑j∈N‍Γj∗Λjx0.Proof.
Since{Λj∈B(H,Hj):j∈N} and {Γj∈B(H,Hj):j∈N}are a pair of pseudo g-frames, for any y∈H, we have(30)y,x0=∑j∈N‍Λjy,Γjx0=y,∑j∈N‍Λj∗Γjx0.So x0=∑j∈N‍Λj∗Γjx0. It follows that x0=∑j∈N‍Λj∗Γjx0=∑j∈N‍Γj∗Λjx0 by Theorem 20.Theorem 22.
Suppose that{θj∈B(H,Hj):j∈N} is a g-orthonormal basis for H and T1 and T2 are g-preframe operators associated with g-frames {Λj∈B(H,Hj):j∈N} and {Γj∈B(H,Hj):j∈N}, respectively. Then, {Λj:j∈N} and {Γj:j∈N} are g-biorthogonal if and only if T1∗T2=I, where I is the identity operator in B(H).Proof.
SinceT1 and T2 are g-preframe operators associated with g-frames {Λj:j∈N} and {Γj:j∈N}, respectively, Λj=θjT1∗ and Γj=θjT2∗ for any j∈N. So, for any i,j∈N and any gi∈Hi, gj∈Hj, we have(31)Λi∗gi,Γj∗gj=T1θi∗gi,T2θj∗gj=θi∗gi,T1T2∗θj∗gj.If {Λj:j∈N} and {Γj:j∈N} are g-biorthogonal, then(32)Λi∗gi,Γj∗gj=δi,jgi,gj=θi∗gi,θj∗gj,kkkkikkkk∀i,j∈N,∀gi∈Hi,gj∈Hj.So(33)θi∗gi,T1T2∗θj∗gj=θi∗gi,θj∗gj,ii∀i,j∈N,∀gi∈Hi,gj∈Hj.It implies that T1T2∗=I.
Conversely, ifT1T2∗=I, then(34)〈Λi∗gi,Γj∗gj〉=〈θi∗gi,T1T2∗θj∗gj〉=〈θi∗gi,θj∗gj〉=δi,j〈gi,gj〉,kkiikkk∀i,j∈N,∀gi∈Hi,gj∈Hj.So {Λj:j∈N} and {Γj:j∈N} are g-biorthogonal.



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