ARCHIVUM MATHEMATICUM (BRNO)
Tomus 46 (2010), 203–209
A NOTE ON FUSION BANACH FRAMES
S. K. Kaushik and Varinder Kumar
Abstract. For a fusion Banach frame ({Gn , vn }, S) for a Banach
∗ (E ∗ ), v ∗ }, T ) is a fusion Banach frame for E ∗ , then
space E, if ({vn
n
∗ (E ∗ ), v ∗ }, T ) is called a fusion bi-Banach frame for E. It
({Gn , vn }, S; {vn
n
is proved that if E has an atomic decomposition, then E also has a fusion
bi-Banach frame. Also, a sufficient condition for the existence of a fusion
bi-Banach frame is given. Finally, a characterization of fusion bi-Banach
frames is given.
1. Introduction
Frames for Hilbert spaces were introduced by Duffin and Schaeffer [5] in 1952
and re-introduced in 1986 by Daubechies, Grossmann and Meyer [4]. Casazza [2]
and Benedetto and Fickus [1] have studied frames in finite dimensional spaces
which attracted more attention due to their use in signal processing. Frames are
now used as a tool in many areas like data compression, sampling theory, optics,
filter banks, signal detection, time-frequency analysis etc.
The concept of frames in Hilbert spaces was extended to Banach spaces by
Feichtinger and Gröchenig [6] who introduced the concept of atomic decompositions
in Banach spaces. This concept was further generalized by Gröchenig [7] who
introduced the notion of Banach frames for Banach spaces. Jain et al. [9], generalized
Banach frames in Banach spaces and introduced frames of subspaces (Fusion Banach
frames) for Banach spaces. They gave the following definition of a fusion Banach
frame.
Definition 1.1 ([9]). Let E be a Banach space. Let {Gn } be a sequence of
non-trivial subspaces of E and {vn } be a sequence of bounded linear projections
such that vn (E) = Gn , n ∈ N. We associate a Banach space A and an operator
S : A → E with the space E. Then ({Gn , vn }, S) is called a frame of subspaces
(fusion Banach frame) for E with respect to A if
(i) {vn (x)} ∈ A, for all x ∈ E,
(ii) there exist constants A, B (0 < A ≤ B < ∞) such that
AkxkE ≤ k{vn (x)}kA ≤ BkxkE ,
x∈E,
2000 Mathematics Subject Classification: primary 42C15; secondary 42A38.
Key words and phrases: atomic decompositions, fusion Banach frames, fusion bi-Banach frames.
Received September 24, 2009, revised April 2010. Editor V. Müller.
204
S. K. KAUSHIK AND V. KUMAR
(iii) S is a bounded linear operator such that
S({vn (x)}) = x ,
x∈E.
The following lemma, proved in [9], is used in the sequel
Lemma 1.2. Let {Gn } be a sequence of non-trivial subspaces of E and {vn } be a
sequence of bounded linear projections with vn (E) = Gn , n ∈ N. If {vn } is total
over E, i.e., {x ∈ E : vn (x) = 0, for all n ∈ N} = {0}, then A = {{vn (x)} : x ∈ E}
is a Banach space with norm k{vn (x)}kA = kxkE , x ∈ E.
For other related notions on frames in Banach spaces one may refer to
[3, 8, 10, 11].
In the present paper, we introduce fusion bi-Banach frames for a Banach space
E. We prove that if E has an atomic decomposition, then E also has a fusion
bi-Banach frame. Also, a sufficient condition for the existence of fusion bi-Banach
frames is given. Finally, a characterization of fusion bi-Banach frames is obtained.
2. Main Results
One may observe that, if ({Gn , vn }, S) is a fusion Banach frame for E with
respect to some associated Banach space A, then there may not exist a Banach
space A1 associated with E ∗ together with an operator T : A1 → E ∗ such that
({vn∗ (E ∗ ), vn∗ }, T ) is a fusion Banach frame for E ∗ with respect to A1 .
In this regard, we have the following examples
Example 2.1. Consider the Banach space
E = `∞ (X) = {xn } : xn ∈ X; sup kxn kX < ∞
1≤n<∞
equipped with the norm k{xn }kE =
sup kxn kX , {xn } ∈ E, where (X, k · k) is
1≤n<∞
a Banach space. For each n ∈ N, define Gn = {δnx : x ∈ X} and vn (x) = δnxn ,
x = {xn } ∈ E, where δnx = (0, 0, . . . , 0, x , 0, . . .) for all n ∈ N and x ∈ X. Then
↓
n-th place
by Lemma 1.2, there exist an associated Banach space A = {{vn (x)} : x ∈ E}
with norm k{vn (x)}kA = kxkE , x ∈ E together with an operator S : A → E given
by S({vn (x)}) = x, x ∈ E such that ({Gn , vn }, S) is a fusion Banach frame for E
with respect to A. But, there does not exist a Banach space A1 associated with
E ∗ together with an operator T : A1 → E ∗ such that ({vn∗ (E ∗ ), vn∗ }, T ) is a fusion
∞
S
Banach frame for E ∗ with respect to A1 . For otherwise,
Gn = E, which is
n=1
not true.
Example 2.2. Let E be a Banach space defined as
E = c0 (X) = {xn } : xn ∈ X; lim kxn kX = 0
n→∞
equipped with the norm given by
k{xn }kE = sup kxn kX ,
1≤n<∞
where (X, k · k) is a Banach space.
A NOTE ON FUSION BANACH FRAMES
205
Define a sequence {Gn } of subspaces of E by
x
x
G2n−1 = {δ2n−1
− 2n−1 δ2n
: x ∈ X}
x
G2n = {δ2n
: x ∈ X} .
Also define operators vn on E by
x
x
2n−1
v2n−1 (x) = δ2n−1
− 2n−1 δ2n2n−1
2n−1 x2n−1 +x2n
v2n (x) = δ2n
for all x = {xn } ∈ E and n ∈ N .
Then by Lemma 1.2 there exist an associated Banach space A and an operator
S : A → E such that ({Gn , vn }, S) is a fusion Banach frame for E with respect to
A.
∞
S
If
Gn 6= E, then there exists 0 6= f = {fi } ∈ E ∗ such that f (y) = 0
n=1
for all y ∈ Gn , n ∈ N. This would imply fn = 0 for all n ∈ N and hence f = 0.
Therefore, by Lemma 1.2 again, there exist a Banach space A1 associated to E ∗
and an operator T : A1 → E ∗ such that ({vn∗ (E ∗ ), vn∗ }, T ) is a fusion Banach frame
for E ∗ with respect to A1 .
In view of the above discussion, we define the following
Definition 2.3. Let E be a Banach space. Let {Gn } be a sequence of non-trivial
subspaces of E and {vn } be a sequence of bounded linear projections such that
vn (E) = Gn , n ∈ N. If there exist Banach spaces A and A1 associated with
E and E ∗ respectively and operators S : A → E and T : A1 → E ∗ such that
({Gn , vn }, S) is a fusion Banach frame for E with respect to A and ({vn∗ (E ∗ ), vn∗ }, T )
is a fusion Banach frame for E ∗ with respect to A1 , then we call the system
({Gn , vn }, S; {vn∗ (E ∗ ), vn∗ }, T ) a fusion bi-Banach frame for E
In view of Remark 3.2.1 in [9], we have
Every reflexive Banach space has a fusion bi-Banach frame.
Recall that if E is a Banach space and Ed is an associated Banach space of
scalar-valued sequences, indexed by N, {xn } is a sequence in E and {fn } is a
sequence in E ∗ , then the pair ({fn }, {xn }) is called an atomic decomposition for E
with respect to Ed if
(i) {fn (x)} ∈ Ed , x ∈ E;
(ii) there exist constants A, B with 0 < A ≤ B < ∞ such that
AkxkE ≤ k{fn (x)}kEd ≤ BkxkE ,
(iii) x =
∞
P
x∈E;
fn (x)xn , x ∈ E.
n=1
The next result is regarding the existence of fusion bi-Banach frames for a
Banach space having an atomic decomposition.
Theorem 2.4. Let E be a Banach space. If E has an atomic decomposition, then
it also has a fusion bi-Banach frame.
206
S. K. KAUSHIK AND V. KUMAR
Proof. Let ({fn }, {xn }) be an atomic decomposition for E with respect to Ed .
Define Gn = [xn ], n ∈ N and vn (x) = fn (x)xn , n ∈ N. Then there exist an
associated Banach space A = {{vn (x)} : x ∈ E} together with an operator
S : A → E such that ({Gn , vn }, S) is a fusion Banach frame for E with respect
∞
S
Gn = E (as [xn ] = E). So, vn∗ (f ) = 0 for all n ∈ N imply
to A. Further
n=1
f = 0, where f ∈ E ∗ . Thus, {vn∗ } is total over E ∗ and so by Lemma 1.2, there
exist an associated Banach space A1 and an operator T : A1 → E ∗ such that
({vn∗ (E ∗ ), vn∗ }, T ) is a fusion Banach frame for E ∗ with respect to A1 . Hence,
({Gn , vn }, S; {vn∗ (E ∗ ), vn∗ }, T ) is a fusion bi-Banach frame for E.
Next, we observe that if E be a Banach space and {Gn } be a sequence of
non-trivial subspaces of E with associated sequence of projections {vn } with
vn (E) = Gn , n ∈ N, then it is possible that there exist a Banach space A1
associated with E ∗ together with a bounded linear operator T : A1 → E ∗ such that
({vn∗ (E ∗ ), vn∗ }, T ) is a fusion Banach frame for E ∗ with respect to A1 and there
may not exist any Banach space A associated with E together with an operator
S : A → E such that ({Gn , vn }, S) is a fusion Banach frame for E with respect to
A. Indeed, let
∞
n
o
X
E = `2 (X) = {xn } : xn ∈ X;
kxn k2X < ∞ ,
n=1
where (X, k · k) is a Banach space, equipped with the norm given by
k{xn }kE =
∞
X
kxn k2X
1/2
.
n=1
x
x
n+1
x
Define for n ∈ N, Gn = {δ1x + δn+1
: x ∈ X} and vn (x) = δ1 n+1 + δn+1
,
x
x = {xn } ∈ E, where δn = (0, 0, . . . , 0, x , 0, . . .), x ∈ X.
↓
nth place
Then
∞
S
Gn = E and vi vj = 0 for all i 6= j.
n=1
But, since for any 0 6= x ∈ X, δ1x = (x, 0, 0, . . .) ∈ E is such that vn (δ1x ) = 0, for
all n ∈ N , there exist no associated Banach space A such that ({Gn , vn }, S) is a
fusion Banach frame for E with respect to A. However, there exist a Banach space
A0 and an operator T : A0 → E ∗ such that ({vn∗ (E ∗ ), vn∗ }, T ) is a fusion Banach
frame for E ∗ with respect to A0 .
In view of the above discussion, we prove the following result
Theorem 2.5. Let E be a Banach space and {Gn } be a sequence of subspaces
∞
S
of E with
Gn = E. Let {vn } be a sequence of projections on E satisfying
n=1
vn (E) = Gn , n ∈ N and vi vj = 0 for all i 6= j. Then there exist Banach spaces
A and A1 associated with E and E ∗ , respectively, and operators S : A → E and
T : A1 → E ∗ such that ({Gn , vn }, S; {vn∗ (E ∗ ), vn∗ }, T ) is a fusion bi-Banach frame
A NOTE ON FUSION BANACH FRAMES
207
for E if every sequence {xn } ⊂ E such that xn ∈ Gn and xn 6= 0, n ∈ N satisfies
∞
T
[xn+1 , xn+2 , . . .] = {0}.
n=1
Proof. Since
∞
S
Gn = E, there exist an associated Banach space A1 and
n=1
a bounded linear operator T : A1 → E ∗ such that ({vn∗ (E ∗ ), vn∗ }, T ) is a fusion
Banach frame for E ∗ with respect to A1 . Let, if possible, there exist no Banach
space A associated with E such that ({Gn , vn }, S) is a fusion Banach frame for
E with respect to A where S : A → E is a bounded linear operator. Now, since
∞
n
S
P
Gn = E and vi vj = 0 for all i 6= j, un =
vi is a bounded linear projection
n=1
i=1
h
i
∞
n
n
∞
S
S
S
S
of E onto
Gi along
Gi , n ∈ N. Write E =
Gi ⊕
Gi ,
i=1
i=n+1
i=1
i=n+1
n ∈ N. Then
{x ∈ E : vi (x) = 0, i = 1, 2, . . . , n} =
∞
h [
i
Gi ,
n ∈ N.
i=n+1
Since ({Gn , vn }, S) is not a fusion Banach frame for E with respect to any
∞ S
∞
T
associated Banach space, there exists 0 6= x ∈
Gi . So, there exists
n=1
y1 =
m
P1
zi
i=1
m
S1
dist x,
y2 =
i=n+1
where zi ∈ Gi (1 ≤ i ≤ m1 ) such that dist(x, y1 ) < 1, that is,
Gi
i=1
m
P2
< 1. Also, x ∈
∞
S
Gi . So, we can choose m2 > m1 and
i=m1 +1
zi , where zi ∈ Gi (m1 + 1 ≤ i ≤ m2 ) such that dist x,
i=m1 +1
m
S2
Gi
<
i=m1 +1
1
. Proceeding like this, for each n ∈ N, we get a sequence {zn } ⊂ E and an
2
increasing sequence {mn } of positive integers such that zn ∈ Gn , n ∈ N and
m
Sn
1
dist x,
Gi < .
n
i=mn−1 +1
Thus x ∈ [zn+1 , zn+2 , . . . ], n ∈ N. Consider a sequence {xn } ⊂ E with 0 6= xn ∈ Gn ,
n ∈ N such that xn = zn whenever zn 6= 0. Then x ∈ [xn+1 , xn+2 , . . . ], n ∈ N.
∞
T
Hence
[xn+1 , xn+2 , . . . ] 6= {0}.
n=1
Finally, we give a characterization of fusion bi-Banach frames in terms of a
sequence in bv0 , where bv0 is the linear space of all sequences {αn } of scalars with
∞
P
lim αn = 0 and for which the norm k{αn }k =
|αn+1 − αn | is finite.
n→∞
n=1
Theorem 2.6. Let E be a Banach space and ({Gn , vn }, S) be a fusion Banach
frame for E, where the projections {vn } on E are such that vi vj = 0 for all i 6= j.
Then ({Gn , vn }, S; {vn∗ (E ∗ ), vn∗ }, T ) is a fusion bi-Banach frame for E if and only
208
S. K. KAUSHIK AND V. KUMAR
if for every x ∈ E, there exist {αj } ∈ bv0 and z ∈ E such that vn (x) = αn vn (z),
n
P
n ∈ N and sup vi (z) < ∞.
1≤n<∞
i=1
Proof. Let ({Gn , vn }, S; {vn∗ (E ∗ ), vn∗ }, T ) be a fusion bi-Banach frame for E. For
k
P
each k ∈ N, write uk =
vi . Then lim uk (x) = x, x ∈ E. Therefore, there exists
k→∞
i=1
a sequence {mn } of positive integers such that
kx − uk (x)k <
m
Pn
1
4n+1
,
k ≥ mn ,
n ∈ N.
2
, n ∈ N.
n
4
i=mn−1 +1
∞
∞
∞
P
P
P
So,
2n−1 kyn k ≤
2−n . Thus, the series
2n−1 yn converges.
Take yn =
n=1
Put z =
∞
P
vi (x), n ∈ N. Then kyn k ≤
n=1
2
n−1
n=1
1−n
yn and αj = 2
, mn−1 + 1 ≤ j ≤ mn , n ∈ N.
n=1
Therefore, {αj } ∈ bv0 . Also, we have
vj (z) = 2n−1 vj (x) ,
mn−1 + 1 ≤ j ≤ mn , n ∈ N .
Hence, vj (x) = αj vj (z), j ∈ N.
Conversely, for integers p < q, we have
q
q
i−1
i
X
X
X
X
vj (z) αi
vj (z) −
vi (x) = i=p
i=p
j=1
j=1
q−1
n
X
X
vj (z)
≤ |αp | +
|αi − αi+1 | + |αq | sup 1≤n<∞
i=p
Since, {αj } ∈ bv0 ,
n
P
j=1
vi (x) is a Cauchy sequence and hence converges.
i=1
Also, since {vn } is total on E and
n
X
vj x − lim
vi (x) = 0 ,
n→∞
it follows that x = lim
n
P
n→∞ i=1
for all
j ∈ N,
i=1
vi (x). Therefore,
h S
∞
n=1
i
Gi = E. Thus, {vn∗ } is total
over E ∗ and so by Lemma 1.2, there exist a Banach space A1 associated with
E ∗ and an operator T : A1 → E ∗ such that ({vn∗ (E ∗ ), vn∗ }, T ) is a fusion Banach
frame for E ∗ with respect to A1 . Hence, ({Gn , vn }, S; {vn∗ (E ∗ ), vn∗ }, T ) is a fusion
bi-Banach frame for E.
Acknowledgement. The authors thank the referees for their useful suggestions
towards the improvement of the paper. The research of second author is supported
by the CSIR (India) (vide letter File No. 09/045(0647)/2006-EMR-I).
A NOTE ON FUSION BANACH FRAMES
209
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Department of Mathematics, Kirori Mal College
University of Delhi, Delhi 110007, India
E-mail: shikk2003@yahoo.co.in
Department of Mathematics, University of Delhi
Delhi 110007, India
E-mail: vicky.h1729@gmail.com