CC BY
17152 Probl. Anal. Issues Anal. Vol. 9 (27), No 2, 2020, pp. 152-162
DOI: 10.15393/j3.art.2020.8630
UDC 517.98, 517.521
R. Zarghami Farfar, V. Sadri, R. Ahmadi
SOME IDENTITIES AND INEQUALITIES FOR G-FUSION
FRAMES
Abstract. G-fusion frames, which are obtained from the combination of g-frames and fusion frames, were recently introduced for Hilbert spaces. In this paper, we present a new identity for g-frames, which was given by Najati for a special case. Also, by using the idea of this identity and the dual frames, some equalities and inequalities are presented for g-fusion frames.
Key words: g-frame, dual g-frame, g-fusion frame, dual g-fusion frame
2010 Mathematical Subject Classification: Primary 42C15; Secondary 46C99, 41A58.
1. Introduction. Recent developments in the frame theory and their applications are the result of some mathematicians' efforts in this topic (see [10], [13], [12], [3], [6], [8]). By more than half a century, this theory has got interesting applications in different branches of science, such as the filter bank theory, signal and image processing, wireless communications, atomic systems, and the Kadison-Singer problem. In 2005, Balan, Casazza, and others found some useful identities for frames by studying properties of the Parseval frames [2]. Simil ar results for fusion frames, g-frames, and K-frames are presented in [18], [21], [1]. In [22], a special kind of frames called g-fusion frames is introduced; they are combinations of g-frames and fusion frames. We present some identities for these frames.
2. Preliminaries. Throughout this paper, H and K are separable Hilbert spaces, is the orthogonal projection from H onto a closed subspace V C H, and B(H, K) is the collection of all the bounded linear operators of H into K. If K = H, then B(H, H) will be denoted by B(H). Also, [Hjis a sequence of Hilbert spaces and Aj E B(H, Hj) for each
© Petrozavodsk State University, 2020
j E J, where J is a subset of Z. The following lemmas from the operator theory will be needed.
Lemma 1. [13] Let V C H be a closed subspace, and T be a linear bounded operator on H. Then
rjl* rjl*
■Kv ± = ftv 1 ftTV-
Lemma 2. [21] Let u E B(H) be adjoint and v := au2 + bu + c where a,b,c E R.
(I) If a > 0, then
inf (vfj>> .
\\f ||=1W'J>> 4a
(II) If a < 0, then
4(xc — b2
sup (vf,f> < —--.
\\f\\=1 4a
Lemma 3. [2] If u,v are operators on H satisfying u + v = idu, then
22 u — v = u — v2.
We define the space H2 := ( Y1 ®Hj)e by
je J ^^ on2
H2 = {{f,}jej : fj E Hj, ^llf)||2< to},
je J
with the inner product defined by
<{/;}, {93}> = £</;>-
je J
It is clear that H2 is a Hilbert space with pointwise operations.
Definition 1. [23] We call the sequence {Aj}jej a g-frame for H with respect to {Hj}jej if there exist 0 < A < B < to, such that for each f E H
WII2< £l|A;f ll2< B
je J
If A = B = 1, we call {A^ }jej a Parseval g-frame. The synthesis and analysis operators in g-frames are defined by
T : H2 —► H , T * : H —► H2
2
T(U,>i€i) = 5>jf3 , T*(f) = [Ajf}i€j.
jes
Therefore, the g-frame operator is defined by
Sf = TT *f = ^ A**Aj f.
jes
The operator S is bounded, positive, and invertible. If Aj := AjS-1, then [Aj }j€j is called a (canonical) dual g-frame of [Aj }j€j, and we can write
f = E A **Aj f = E A*A, f. (2)
jes jes
If [Aj}j€j is a g-frame for H with bounds A and B, respectively, then {Aj }j€j is also a g-frame for H with bounds B-1 and A-1, respectively.
Definition 2. [22] Let W = [Wj}j€j be a family of closed subspaces of H, [vj}j€j be a family of weights, i. e., Vj > 0. We say A := (Wj, Aj ,Vj) is a g-fusion frame for H if there exist 0 < A < B < to, such that for each I e H
m ii2< £ v2\\Ai f ii2 < b uii2 . (3)
j€S
It is easy to see that these frames are extensions of g-frames. We call A a Parseval g-fusion frame if A = B = 1. When the right-hand part of (3) holds, A is called a g-fusion Bessel sequence for H with the bound B. Throughout this paper, A is a triple (Wj, Aj,Vj) with j e J.
The synthesis and analysis operators in the g-fusion frames are defined by (for more details, we refer [22])
TA : H2 —► H, Tj : H —► H2
TA ([fj }iej) = Y1 vi A*h, TA(/) = [vj Aj^ f W
j€J
Thus, the g-fusion frame operator is given by
S\f = TAT*f = ^2 A*jAj KWj f. jes
Therefore,
A idn < sA < B idn.
This means that Sa is a bounded, positive, and invertible operator (with an adjoint inverse), and we have
B-lxdH < S-1 < A-1idH.
So, we have the following reconstruction formula for any f e H:
f = tf^Wj AjAj ftWj s-lf = vj A* Ai ftWj f. (4)
jeJ jeJ
Let A := (S—lWj, Ajnw.S-l,Vj). Then A is called the (canonical) dual g-fusion frame of A. Hence, for each f e H we get
f = Y^ tf^WjAj Ai ^Wj f = Y1 tf AjAi ^ f, (5)
jeJ jeJ
where Wj := S-lWj , Aj := AjnWjS-1. Thus, we obtain
(s-lf,f > = £ "HA f \\2. (6)
ieJ
3. The Main Results. Let {Ajbe a g-frame for H with respect to {Hj}jeJ with bounds A,B and {Aj}eJ be a (canonical) dual g-frame of {Aj }jeJ. Suppose that I C J and let
Si : H ^ H
Sif := £ AjA, f. jei
This is a general case of the operator Sj presented in [21]. We have \\Sif ||2= ( sup KSif,h>l)2 = sup f, Aj h>l)2 <
< 1A j f\\2x sup [Aj h\\2< BA~l\\f \\2.
i M=lT
Thus, Si e B(H) and is positive. From (2) we obtain that Si + Sic = idH. Theorem 1. For f e H, we have
£(A;f, A,f>-\\Sif \\2= £ (A,f, A,f> - Hflcf \\2
jei jeic
where F is the complement of I. Proof. For each f e H, we have
2
E(A3 f, A, f > - II E A*Aj f = (Sif, f > -
3J13
jei je
= (SifJ > - (S*Sif,f > = ((ids - SiYSifJ > = (S*c (^ - 5ic )f,f >
= (5* f, f > - (5*Sic f, f > = (/, Sic f > - (Sic f, Si. f > =
£<A; f, A f > - IE A^A, f II2= E (Ai f, A3 f > - IE a,*A.
jeic jeic jeic jeic
and the proof is complete. □
Now, if [Ajis a Parseval g-frame, then Aj = Aj, and we obtain the following famous formula presented in [21]:
ei a f ii2-usif ii2= ei a f 112-ii^ f 112,
jei jei°
where Sif = ejei A*A,-f.
The same can be obtained for g-fusion frames. Let A be a g-fusion frame for H with a (canonical) dual g-fusion frame A = (Wj, Aj ,Vj), where Wj = SaWj and A j = Aj nWj S-1. For simplicity, we denote the following operator with the same symbol Si, where, again, i is a finite subset of J:
Sif = E ^W A* A j f, (Vf e H). (7)
jei
It is easy to check that Si e B(H) and positive. Again, we have
Si + Sic = ids.
Remark 1. Let A be a Parseval g-fusion frame for H. Since B(H) is a C*-algebra and Si is positive, so r(Si) = ||Si|, where r is the spectral radius. Thus
max |A|= r(Si) < 1
Xea(St)
and we conclude that a(Si) e [0,1]. Theorem 2. Let f e H; then
E^2(A^W. f, A j^Wj f > - || Si f ||2= £ v23(A^Wj f, A j-KWj f > - II Sic „
jei jei
2
Proof. The proof follows a similar argument as in the proof of Theorem 1. □
Corollary 1. Let A be a Parseval g-fusion frame for H. Then
^v?\\AjftWj f \\2-1| ^v^w3AjAjftWj f jei jei
^ v* \ \ A.j ftW3 f \ \2 - E v2'Kwj AjAj KW3 f
$ \ \ A3 f \ \ #
jeic jei
Moreover,
E $ \ \ A ^ f \ \ 2+|^ v2nw3 AjAj ftw, f
jei jei
2> 3
Proof. If f e H, we obtain
£ «J \\ A,ftw, f \\2+ \\ 5icf \\2= ((Si + )f, f>
jei
= ((Si + idH - 2Si + S2)f, f > = ((idH - Si + S2)f, f >. Now, by Lemma 2 for a =1, b = -1, and c =1, the inequality holds. □ Corollary 2. Let A be a Parseval g-fusion frame for H. Then
0 < Si - SH < 1 idH
Proof. We have SfSfc = S^ Sf. Then 0 < SfSfc = Sf - S2. Also, by Lemma 2, we get
Si - S2 < 4idH.
The proof is complete. □
Theorem 3. Let A be a g-fusion frame with the g-fusion frame operator Sa. If I C J and f e H, then
£ \ \ A, f \ \ 2+ \ \ 2 % c f \ \ 2= £ V2 \ \ Aj f \ \ 2+\\ 1S!/\\ 2.
jei jeic
Proof. Let Oj := AjnWjS-2 and Xj := 1 Wj. Example 2.2 [22] shows that (Xj, Oj,Vj) is a Parseval g-fusion frame for H. Then, by Corollary 1, we have
2
E и0з*ъ f il2- Il £ ^0**0з^ f
jei jei
£ vj ii0j^ f il2-1E v3^f
jeic jei
By replacing f by S I f and the fact that
vjnX] 0j 0j nxj f = £ v3(03 )0j
jei jei
0j0jf = £ vK0i)*0j*Xi f =
jei
_л _ 1 _ 1
£ vj (Л3S- 2 nXj У лЗ KWj sл 2 nxj f jei
V-Л —1 —1
vjsA 2 -kw. Л* Aj s- 2 f =
jei
_ С 2 С С 2 f
= oA OiOA J ,
the proof is complete. □
Corollary 1. Let A be a g-fusion frame with the g-fusion frame operator SA .If i C J, then
1 4
Proof. In the proof of Theorem 3, we showed that
_ i i
Uj nxj 0j * - Q
0 < Si — SiS-1Si < — Sa.
^ 0**03 nx, f = s-1 SiS-1 f.
jei
By Corollary 2, we get
0 < £v3^x,0**0j^f — (£0**0зKxj/)2 < 1 ldH. jei jei
Therefore,
-1 1 -1 1 0 < S- 2 (Si — SiS- Si)S-2 < 4idH
and the proof is complete. □
Corollary 2. Suppose that Л is a g-fusion frame with the g-fusion frame operator Sa . If i Ç J and f G H, then
1 3
£ f ||2+|S-1 Sic f ||2> - lis-1!"
jei
2
2
Proof. By Theorem 3 and Corollary 1, we can write
£ v? 11 Ajf 11 2+ 11 1S* f 11 =
jei
J>f 11 O nXj SI f I1 2+ I I S1 f I1 ^
jei jev
3 1 3 3
> 3 11 ^ f 11 2=4 (SAf,f >> 3 11 S-1 11 -1 11 f 112.
The poof is complete. □
Theorem 4. Let A be a Parseval g-fusion frame for H and i C J. Then (I) 0 < Si — Sf < 4 ids.
1 3
(II) 1 < sf + Sfc < 2idH.
Proof. (I) Since Si + Sic = idu, SiSic + S2c = Sic. Thus,
Si Sic = Sic — S^ = Sic (idn — Sic) = Sic Si-
But A is Parseval, so 0 < SiSic = Si — S2. On the other hand, by Lemma 3, we get
Si — S2 < 4idn -(II) We have seen that SjSjc = SjcSj; then, by Lemma 3,
S2 + S2 = idn — = 2S2 — 2 Si + idn >2 ^h-
On the other hand, we have, again, by Lemma 3 and 0 < Sf — Sf:
3
s2 + 4 < %dH + 2^ — 2s2 < 3idH.
This completes the proof. □
Corollary 1. Let A be a g-fusion frame with the g-fusion frame operator SA. If i C J, then
1 1 1 3
2Sa < SiSA Si — SicSA Sic < 2
Proof. We have
Y^ tf OO? = ^ 2 SiSA 2 f.
jei
Therefore, similarly to the proof of Corollary 1, we get, by Theorem 4, item (II),
1 -1 -1 2 -1 -1 2 3
2id h < (S-2 SiS- 2) + (S-2 SicS- 2) < 2
and the proof is now evident. □
Theorem 5. Let A be a g-fusion frame with the g-fusion frame operator S- .If i C J, then, for any f e H,
: $ \ \ a, f \ \2- V v2 \ \ A, *w, Mif \ \ 2=
/ , Uj \ \ li3'>wjJ\ \ Vj\ \ ^J'
jei jei
\ \ A j-KWj f \ \ 2-Y, v2 \ \ A^w3 Mic f \ \2,
jeic jei
where
Mif = £ v2 nwj A*A^w3 f.
e
Proof. Via the definition of S-, it is clear that Mi + Mic = S-. Therefore, S-lMi + S-lMic = idH. Hence, by Lemma 3
S-lMi - S-lMic = (S-lMi)2 - (S-lMic)2.
Thus, for each f,g e H we obtain
(S-lMif,g> - (S-lMiS-lMif,g> = (S-lMicf,g> - (S-lMicS-lMicf,g>. We choose g to be g = SAf, and we can get
(Mif, f > - (S-lMif, Mif > = (Micf, f> - (S~-lMicf, Micf >. Finally, by (6), the proof is complete. □
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Received May 27, 2019.
In revised form, November 29, 2019.
Accepted January 11, 2020.
Published online April 29, 2020.
Ramazan Zarghami Farfar
University of Tabriz
Marand Faculty of Technical and Engineering, Iran.
E-mail: [email protected]
Vahid Sadri
Department of Mathematics, Faculty of Tabriz Branch,
Technical and Vocational University (TVU), East Azarbaijan, Iran.
E-mail: [email protected]
Reza Ahmadi
University of Tabriz
Institute of Fundamental Sciences, Iran.
E-mail: [email protected]
