on ð®âbanach bessel sequences in banach spaces · optics, filter banks, signal detection as...
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Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 9 (2017), pp. 5129-5142
© Research India Publications
http://www.ripublication.com
On ð® âBanach Bessel Sequences in Banach Spaces
Ghanshyam Singh Rathore1 and Tripti Mittal2
1Department of Mathematics and Statistics, University College of Science,
M.L.S. University, Udaipur, INDIA.
2Department of Mathematics, Northern India Engineering College,
G.G.S. Inderprastha University, Delhi, INDIA.
Abstract
Abdollahpour et.al [1] generalized the concepts of frames for Banach Spaces
and defined ð âBanach frames in Banach spaces. In the present paper, we
define ð â Banach Bessel sequences in Banach spaces and study their
relationship with ð âBanach frames. A sufficient condition for a sequence of
bounded operators to be a ð âBanach frame, in terms of ð âBanach Bessel
sequence has been given. Further, we study bounded linear operators on an
associated Banach space ðð. Also, a necessary and sufficient condition for a
kind of ð âBanach Bessel sequence to be a ð âBanach frame has been given.
Furthermore, we give stability condition for the stability of ð âBanach Bessel
sequences.
1. INTRODUCTION
Frames are main tools for use in signal processing, image processing, data compression
and sampling theory etc. Today even more uses are being found for the theory such as
optics, filter banks, signal detection as well as study of Besov spaces, Banach space
theory etc. Frames for Hilbert spaces were introduced by Duffin and Schaeffer [12].
Later, in 1986, Daubechies, Grossmann and Meyer [11] reintroduced frames and found
a new application to wavelet and Gabor transforms. For a nice introduction to frames,
one may refer [4].
5130 Ghanshyam Singh Rathore and Tripti Mittal
Coifman and Weiss [10] introduced the notion of atomic decomposition for function
spaces. Feichtinger and Gröchening [14] extended the notion of atomic decomposition
to Banach spaces. Gröchening [16] introduced a more general concept for Banach
spaces called Banach frame. He gave the following definition of a Banach frame:
Definition 1.1. ([16]) Let ð be a Banach space and ðð1 an associated Banach space
of scalar-valued sequences indexed by â. Let {ðð} â ðâ and ð: ðð1â ð be given.
Then the pair ({ðð}, ð) is called a Banach frame for ð with respect to ðð1, if
1. {ðð(ð¥)} â ðð1, for each ð¥ â ð.
2. there exist positive constants ðŽ and ðµ with 0 < ðŽ †ðµ < â such that
ðŽâð¥âð †â{ðð(ð¥)}âðð1†ðµâð¥âð, ð¥ â ð. (1.1)
3. ð is a bounded linear operator such that ð({ðð(ð¥)}) = ð¥, ð¥ â ð.
The positive constants ðŽ and ðµ , respectively, are called lower and upper frame
bounds of the Banach frame ({ðð}, ð) . The operator ð: ðð1â ð is called the
reconstruction operator ( or the pre-frame operator ). The inequality (1.1) is called the
frame inequalty.
Banach frames and Banach Bessel sequence in Banach spaces were further studied in
[3, 17, 18, 19, 20].
Over the last decade, various other generalizations of frames for Hilbert spaces have
been introduced and studied. Some of them are bounded quasi-projectors by Fornasier
[15]; Pseudo frames by Li and Ogawa [21]; Oblique frames by Eldar [13]; Christensen
and Eldar [5]; (,;Y )-Operators frames by Cao et.al [2]. W. Sun [24] introduced and
defined ð âframes in Hilbert spaces and observed that bounded quasi-projectors,
frames of subspaces, Pseudo frames and Oblique frames are particular cases of
ð âframes in Hilbert spaces. ðº âframes are further studied in [6, 7, 8, 9, 22, 25].
Abdollahpour et.al [1] generalized the concepts of frames for Banach Spaces and
defined ð â Banach frames in Banach spaces. In the present paper, we define
ð âBanach Bessel sequences in Banach spaces and study their relationship with
ð âBanach frames. A sufficient condition for a sequence of bounded operators to be
a ð âBanach frame, in terms of ð âBanach Bessel sequence has been given. Further,
we study bounded linear operators on an associated Banach space ðð . Also, a
necessary and sufficient condition for a kind of ð âBanach Bessel sequence to be a
ð âBanach frame has been given. Furthermore, we give stability condition for the
stability of ð âBanach Bessel sequences.
On G-Banach Bessel Sequences in Banach Spaces 5131
2. PRELIMINARIES
Throughout this paper, ð will denote a Banach space over the scalar field ð (â or
â), ðð is an associated Banach space of vector valued sequences indexed by â. ð» is
a separable Hilbert space and ð (ð, ð») is the collection of all bounded linear operators
from ð into ð».
A Banach space of vector valued sequences (or BV-space) is a linear space of
sequences with a norm which makes it a Banach space. Let ð be a Banach space and
1 < ð < â, then
ð = {{ð¥ð}: ð¥ð â ð; (â âð¥ðâððââ )
1
ð < â}
and
ââ = {{ð¥ð}: ð¥ð â ð; supðââ
âð¥ðâ < â}
are BV space of ð.
Definition 2.1. ([1]) Let ð be a Banach space and ð» be a separable Hilbert space.
Let ðð be an associated Banach space of vector-valued sequences indexed by â. Let
{Îð}ðââ â ð (ð, ð») and ð: ðð â ð be given. Then the pair ({Îð}, ð) is called a
ð âBanach frame for ð with respect to ð» and ðð, if
1. {Îð(ð¥)} â ðð, for each ð¥ â ð.
2. there exist positive constants ðŽ and ðµ with 0 < ðŽ †ðµ < â such that
ðŽâð¥âð †â{Îð(ð¥)}âðð†ðµâð¥âð, ð¥ â ð. (2.1)
3. ð is a bounded linear operator such that ð({Îð(ð¥)}) = ð¥, ð¥ â ð.
The positive constants ðŽ and ðµ, respectively, are called the lower and upper frame
bounds of the ð âBanach frame ({Îð}, ð). The operator ð ⶠðð â ð is called the
reconstruction operator and the inequality (2.1) is called the frame inequality for
ð âBanach frame ({În}, ð).
The following lemma, proved in [23], is used in the sequel:
Lemma 2.2. Let ð be a Banach space and ð» be a separable Hilbert space. Let
{Îð} â ð (ð, ð») be a sequence of non-zero operators. If {Îð} is total over ð, i.e.,
5132 Ghanshyam Singh Rathore and Tripti Mittal
{ð¥ â ð: ð¬ð(ð¥) = 0, ð â â} = {0}, then ð = {{Îð(ð¥)}: ð¥ â ð} is a Banach space
with the norm given by â{Îð(ð¥)}âð = âð¥âð , ð¥ â ð.
3. ð® âBANACH BESSEL SEQUENCES
We begin this section with the following definition of ð âBanach Bessel Sequences
in Banach spaces.
Definition 3.1. Let ð be a Banach space and ð» be a separable Hilbert space. Let ðð
be an associated Banach space of vector-valued sequences indexed by â. A sequence
of operators {Îð} â ð (ð, ð») is said to be a ð âBanach Bessel sequence for ð with
respect to ðð, if
1. {Îð(ð¥)} â ðð, for each ð¥ â ð.
2. there exist a positive constant ð such that
â{Îð(ð¥)}âðð†ðâð¥âð , ð¥ â ð. (3.1)
The positive constant ð is called a ð âBanach Besssel bound (or simply, a bound)
for the ð âBanach Bessel sequence {Îð}.
In the view of Definition 3.1, the following observations arise naturally:
(I). If ({Îð}, ð) ({Îð} â ð (ð, ð»), ð ⶠðð â ð) is a ð â Banach frame for ð
with respect to ðð and with bounds ðŽ, ðµ, then {Îð} is a ð âBanach Bessel
sequence for ð with respect to ðð and with bound ðµ.
(II). A sequence {Îð} â ð (ð, ð») is a ð âBanach Bessel sequence for ð with
respect ðð if and only if the coefficient mapping ð ⶠð â ðð given by
ð(ð¥) = {Îð(ð¥)}, ð¥ â ð is bounded.
(III). For each ð â {1,2, ⊠. , ð} , if {Îð,ð} â ð (ð, ð») is a ð â Banach Bessel
sequence for ð with respect to ðð and with bound ðð, then {â ðŒðÎð,ððð=1 } is
also a ð â Banach Bessel sequence for ð with respect to ðð with
bound â |ðŒð|ðððð=1 , where ðŒð â â, ð = 1,2, ⊠, ð.
(IV). Let ({Îð}, ð) ({Îð} â ð (ð, ð»), ð ⶠðð â ð) be a ð âBanach frame for ð
with respect to ðð and with bounds ðŽ , ðµ . Let {Îð} â ð (ð, ð») be a
sequence of operators such that {Îð + Îð} is a ð âBanach Bessel sequence
for ð with respect to ðð and with bound ðŸ. Then {Îð} is also a ð âBanach
Bessel sequence for ð with respect to ðð with bound ðŸ + ðµ.
On G-Banach Bessel Sequences in Banach Spaces 5133
Remark 3.2. The converse of the observation (I) need not be true. In this regard, we
give the following example:
Example 3.3. Let ð» be a separable Hilbert space and let ðž = ðâ(ð») = {{ð¥ð} ⶠð¥ð â
ð»; sup1â€ð<ââð¥ðâð» < â} be a Banach space with the norm given by
â{ð¥ð}âðž = sup1â€ð<â
âð¥ðâð» , {ð¥ð} â ðž
and ðžð â ðâ(ð»).
Now, for each ð â â, define Îð: ðž â ð» by
Î1(ð¥) = ð¿2
ð¥2
Îð(ð¥) = ð¿ð+1ð¥ð+1 , ð > 1
} , ð¥ = {ð¥ð} â ðž,
where, ð¿ðð¥ = (0,0, ⊠, ð¥â
âðð¡âððððð
, 0,0, ⊠), for all ð â â and ð¥ â ð».
Then {Îð(ð¥)} â ðžð , for all ð¥ â ðž and â{Îð(ð¥)}âðžð†âð¥âðž , ð¥ â ðž . Therefore,
{Îð} is a ð âBanach Bessel sequence for ðž with respect to ðžð and with bound 1.
But, there exists no reconstruction operator ð ⶠðžð â ðž such that ({Îð}, ð) is a
ð âBanach frame for ðž with respect to ðžð.
Indeed, if possible, let ({Îð}, ð) be a ð âBanach frame for ðž with respect to ðžð. Let
ðŽ and ðµ be choice of bounds for ({Îð}, ð). Then
ðŽâð¥âðž †â{Îð(ð¥)}âðžð†ðµâð¥âðž , ð¥ â ðž. (3.2)
Let ð¥ = (1,0,0, . . ,0, . . ) be a non-zero element in ðž such that Îð(ð¥) = 0, for all ð â
â. Then, by frame inequality (3.2), we get ð¥ = 0. Which is a contradiction.
Remark 3.4. In view of observation (IV), there exists, in general, no reconstruction
operator ð0 ⶠðð0â ð such that ({Îð}, ð0) is a ð â Banach frame for ð with
respect to ðð0, where ðð0
is some associated Banach space of vector-valued sequences
indexed by â. Regarding this, we give the following example:
Example 3.5. Let ð» be a separable Hilbert space and let ðž = ðâ(ð») = {{ð¥ð} ⶠð¥ð â
ð»; sup1â€ð<ââð¥ðâð» < â} be a Banach space with the norm given by
â{ð¥ð}âðž = sup1â€ð<â
âð¥ðâð» , {ð¥ð} â ðž.
5134 Ghanshyam Singh Rathore and Tripti Mittal
Now, for each ð â â, define Îð: ðž â ð» by
Îð(ð¥) = ð¿ðð¥ð , ð¥ = {ð¥ð} â ðž,
where, ð¿ðð¥ = (0,0, ⊠, ð¥â
âðð¡âððððð
, 0,0, ⊠), for all ð â â and ð¥ â ð».
Thus {Îð} is total over ðž . Therefore, by Lemma 2.2, there exists an associated
Banach space ð = {{Îð(ð¥)}: ð¥ â ðž} with the norm given by â{Îð(ð¥)}âð = âð¥âðž ,
ð¥ â ðž.
Define ð ⶠð â ðž by ð({Îð(ð¥)}) = ð¥, ð¥ â ðž . Then ({Îð}, ð) is a ð â Banach
frame for E with respect to ð with bounds ðŽ = ðµ = 1.
Let {Îð} â ð (ðž, ð») be a sequence of operators defined as
Î1(ð¥) = Î2(ð¥),
Îð(ð¥) = Îð(ð¥), ð ⥠2, ð â â} , ð¥ â ðž.
Then, for each ð¥ â ðž, {Îð(ð¥)} â ð and
â{(Îð + Îð)(ð¥)}âð †â{Îð(ð¥)}âð + â{Îð(ð¥)}âð
†â{Îð(ð¥)}âð + â{Îð(ð¥)}âð
= 2â{Îð(ð¥)}âð
= 2âð¥âðž.
Therefore, {Îð + Îð} is a ð âBanach Bessel sequence for ðž with respect ð and
with bound 2. Thus, by observation (IV), {Îð} is a ð âBanach Bessel sequence for
ðž with respect to ð and with bound 3. But, there exists no reconstruction operator
ð0: ð0 â ðž such that ({Îð}, ð0) is a ð âBanach frame for ðž with respect to ð0,
where ð0 is some associated Banach space of vector-valued sequences indexed by
â.
Note. If the Bessel bound of ð âBanach Bessel sequence {Îð + Îð}, ðŸ < âðââ1,
then there exists an associated Banach space ð0 and a reconstruction operator
ð0: ð0 â ð such that ({Îð}, ð0) is a normalized tight ð âBanach frame for ð with
respect to ð0.
In this regard, we give the following result:
Theorem 3.6. Let ({Îð}, ð) ({Îð} â ð (ð, ð»), ð: ðð â ð) be a ð âBanach frame
for ð with respect to ðð. Let {Îð} â ð (ð, ð») be a sequence of operators such that
On G-Banach Bessel Sequences in Banach Spaces 5135
{Îð + Îð} be a ð âBanach Bessel sequence for ð with respect to ðð and with
bound ðŸ < âðââ1. Then, there exists a reconstruction operator ð0: ðð0â ð such that
({Îð}, ð0) is a normalized tight ð âBanach frame for ð with respect to ðð0=
{{Îð(ð¥)}: ð¥ â ð}.
Proof: Let ({Îð}, ð) be a ð âBanach frame for ð with respect to ðð and with
bounds ðŽ, ðµ. Then
ðŽâð¥âð †â{Îð(ð¥)}âðð†ðµâð¥âð , ð¥ â ð.
Since {Îð + Îð} is a ð âBanach Bessel sequence for ð with respect to ðð and with
bound ðŸ, then
(âðââ1 â ðŸ)âð¥âð †â{Îð(ð¥)}âððâ â{(Îð + Îð)(ð¥)}âðð
†â{Îð(ð¥)}âðð, for all ð¥ â ð.
Thus {Îð} is total over ð . Therefore, by Lemma 2.2, there exists an associated
Banach space ðð0= {{Îð(ð¥)}: ð¥ â ð} with the norm given by â{Îð(ð¥)}âðð0
=
âð¥âð , ð¥ â ð. Define ð0: ðð0â ð by ð0({Îð(ð¥)}) = ð¥, ð¥ â ð . Then ð is a
bounded linear operator such that ({Îð}, ð0) is a normalized tight ð âBanach frame
for ð with respect to ðð0.
Towards, the converse of the Theroem 3.6, we give the following result:
Theorem 3.7. Let ({Îð}, ð) and ({Îð}, ð) be two ð âBanach frame for ð with
respect to ðð. Let ð¿ ⶠðð â ðð be a linear homeomorphism such that ð¿({Îð(ð¥)}) ={Îð(ð¥)}, ð¥ â ð . Then {Îð + Îð} is a ð â Banach Bessel sequence for ð with
respect to ðð and with bound
ðŸ = min{âðââðŒ + ð¿â, âðââðŒ + ð¿â1â}
where ð, ð: ðð â ð are the coefficient mappings given by ð(ð¥) = {Îð(ð¥)} and
ð(ð¥) = {Îð(ð¥)}, ð¥ â ð and ðŒ is an identity mapping on ðð.
Proof. For each ð¥ â ð, we have
â{(Îð + Îð)(ð¥)}âðð= â{Îð(ð¥)} + {Îð(ð¥)}âðð
= âðŒ({Îð(ð¥)}) + ð¿({Îð(ð¥)})âðð
†âðŒ + ð¿ââ{Îð(ð¥)}âðð
= âðŒ + ð¿ââð(ð¥)âðð
†(âðŒ + ð¿ââðâ)âð¥âð.
5136 Ghanshyam Singh Rathore and Tripti Mittal
Therefore
â{(Îð + Îð)(ð¥)}âðð†(âðââðŒ + ð¿â)âð¥âð , ð¥ â ð. (3.3)
Similarly, we have
â{(Îð + Îð)(ð¥)}âðð†(âðââðŒ + ð¿â1â)âð¥âð , ð¥ â ð. (3.4)
Using (3.3) and (3.4), it follows that {Îð + Îð} is a ð âBanach Bessel sequence for
ð with respect to ðð and with bound
ðŸ = min{âðââðŒ + ð¿â, âðââðŒ + ð¿â1â}.
Remark 3.8. One may observe that, if ({Îð}, ð) and ({Îð}, ð) are two ð âBanach
frames for ð with respect to ðð, then {Îð + Îð} is a ð âBanach Bessel sequence
for ð with respect to ðð with bound âðâ + âðâ , where ð, ð: ð â ðð are the
coefficient mappings given by ð(ð¥) = {Îð(ð¥)} and ð(ð¥) = {Îð(ð¥)}, ð¥ â ð.
4. BOUNDED LINEAR OPERATORS ON ð¿ð
We begin this section with following result which shows that the action of a bounded
linear operator on ðð to construct a ð âBanach Bessel sequence, using a given
ð âBanach Bessel sequence.
Theorem 4.1. Let {Îð} â ð (ð, ð») be a ð âBanach Bessel sequence for ð with
respect to ðð . Let {Îð} â ð (ð, ð») be such that {Îð(ð¥)} â ðð , ð¥ â ð and let
ð: ðð â ðð be a bounded linear operator such that ð({Îð(ð¥)}) = {Îð(ð¥)}, ð¥ â ð.
Then {Îð} is a ð âBanach Bessel sequence for ð with respect to ðð.
Proof. Let {Îð} be a ð âBanach Bessel sequence for ð with Bessel bound ð. Then,
for each ð¥ â ð, we have
â{Îð(ð¥)}âðð= âð({Îð(ð¥)})âðð
†âðââ{Îð(ð¥)}âðð
†(âðâð)âð¥âð.
Remark 4.2. In the view of Theorem 4.1, we observe that if ({Îð}, ð)
({Îð} â ð (ð, ð»), ð: ðð â ð) is a ð âBanach frame for ð with respect to ðð and
ð: ðð â ðð is a bounded linear operator such that ð({Îð(ð¥)}) = {Îð(ð¥)}, ð¥ â ð.
Then, there exists, in general no reconstruction operator ð: ðð â ðð such that
({Îð}, ð) is a ð âBanach frame for ð with respect to ðð.
In this regarding, we give the following example:
On G-Banach Bessel Sequences in Banach Spaces 5137
Example 4.3. Let ð» be a separable Hilbert space and let ðž = ðâ(ð») = {{ð¥ð} ⶠð¥ð â
ð»; sup1â€ð<ââð¥ðâð» < â} be a Banach space with the norm given by
â{ð¥ð}âðž = sup1â€ð<â
âð¥ðâð» , {ð¥ð} â ðž .
Now, for each ð â â, define Îð: ðž â ð» by
Îð(ð¥) = ð¿ðð¥ð , ð¥ = {ð¥ð} â ðž,
where, ð¿ðð¥ = (0,0, ⊠, ð¥â
âðð¡âððððð
, 0,0, ⊠), for all ð â â and ð¥ â ð».
Thus {Îð} is total over ðž . Therefore, by Lemma 2.2, there exists an associated
Banach space ðžð = {{Îð(ð¥)}: ð¥ â ðž} with the norm given by â{Îð(ð¥)}âðžð= âð¥âðž ,
ð¥ â ðž.
Define ð ⶠðžð â ðž by ð({Îð(ð¥)}) = ð¥, ð¥ â ðž . Then ({Îð}, ð) is a ð â Banach
frame for E with respect to ðžð with bounds ðŽ = ðµ = 1.
Now, define ð: ðžð â ðžð as ð({Îð(ð¥)}) = {Îð(ð¥)}ðâ 1 . Then ð is a bounded
linear operator on ðžð . But, there exists no reconstruction operator ð: ðžð â ðž such
that ({Îð}ðâ 1, ð) is a ð âBanach frame for ðž with respect to ðžð.
Indeed, if possible, let ({Îð}ðâ 1, ð) is a ð âBanach frame for ðž with respect to ðžð.
Let ðŽ and ðµ be choice of bounds for ð âBanach frame ({Îð}ðâ 1, ð). Then
ðŽâð¥âðž †â{Îð(ð¥)}ðâ 1âðžð†ðµâð¥âðž , ð¥ â ðž. (4.1)
Let ð¥ = (1,0,0, ⊠,0, ⊠) be a non-zero element in ðž such that Îð(ð¥) = 0, ð >
1, ð â â. Then, by frame inequality (4.1), we get ð¥ = 0. Which is a contradiction.
In the view of Theorem 4.1 and Example 4.3, we give a necessary and
sufficient condition for a ð âBanach Bessel sequence to be a ð âBanach frame.
Theorem 4.4. Let ({Îð}, ð) ({Îð} â ð (ð, ð»), ð: ðð â ð) be a ð âBanach frame
for ð with respect to ðð . Let {Îð} â ð (ð, ð») be such that {Îð(ð¥)} â ðð , ð¥ â ð.
Let ð: ðð â ðð be a bounded linear operator such that ð({Îð(ð¥)}) = {Îð(ð¥)},
ð¥ â ð. Then, there exists a reconstruction operator ð: ðð â ð such that ({Îð}, ð) is
a ð âBanach frame for ð with respect to ðð if and only if
â{Îð(ð¥)}âðð⥠ðâð¿({Îð(ð¥)})âðð
, ð¥ â ð
where, ðis a positive constant and ð¿: ðð â ðð is a bounded linear operator such that
ð¿({Îð(ð¥)}) = {Îð(ð¥)}, ð¥ â ð.
5138 Ghanshyam Singh Rathore and Tripti Mittal
Proof. Let ðŽÎ and ðµÎ be the frame bounds for the ð âBanach frame ({Îð}, ð) .
Then, for all ð¥ â ð, we have
ððŽÎâð¥âð †ðâ{Îð(ð¥)}âðð
= ðâð¿({Îð(ð¥)})âðð
†â{Îð(ð¥)}âðð
= âð({Îð(ð¥)})âðð
†âðââ{Îð(ð¥)}âðð
†âðâðµÎâð¥âð.
Therefore
(ððŽÎ)âð¥âð †â{Îð(ð¥)}âðð†(âðâðµÎ)âð¥âð, ð¥ â ð.
Put ð = ðð¿. Then ð: ðð â ð is a bounded linear operator such that ð({Îð(ð¥)}) =
ð¥, ð¥ â ð. Hence ({Îð}, ð) is a ð âBanach frame for ð with respect to ðð.
Conversely, let ({Îð}, ð) is a ð âBanach frame for ð with respect to ðð
and with frame bounds ðŽÎ, ðµÎ. Let ð: ð â ðð be the coefficient mapping given by
ð(ð¥) = {Îð(ð¥)}, ð¥ â ð. Then ð¿ = ðð: ðð â ðð is a bounded linear operator such
that ð¿({Îð(ð¥)}) = {Îð(ð¥)}, ð¥ â ð.
Thus, on writing ð = (ðŽÎ
ðµÎ) , where ðµÎ is an upper bound for ð âBanach frame
({Îð}, ð), we have
â{Îð(ð¥)}âðð⥠ðŽÎâð¥âð
= ððµÎâð¥âð
⥠ðâ{Îð(ð¥)}âðð
= ðâð¿({Îð(ð¥)})âðð.
In the following result, we provide a necessary and sufficient condition for a
ð âBanach frame, in terms of ð âBanach Bessel sequence.
Theorem 4.5. Let ({Îð}, ð) ({Îð} â ð (ð, ð»), ð: ðð â ð) be a ð âBanach frame
for ð with respect to ðð. Let {Îð} â ð (ð, ð») be a sequence of operators such that
{Îð + Îð} is a ð âBanach Bessel sequence for ð with respect to ðð and with bound
ðŸ < âðââ1 . Then, there exists a reconstruction operator ð: ðð â ð such that
({Îð}, ð) is a ð âBanach frame for ð with respect to ðð if and only if there exists
a bounded linear operator ð¿: ðð â ðð such that ð¿({Îð(ð¥)}) = {Îð(ð¥)}, ð¥ â ð.
On G-Banach Bessel Sequences in Banach Spaces 5139
Proof. Let ({Îð}, ð) be a ð âBanach frame for ð wih respect to ðð. Let ð: ð â ðð
be the coefficient mapping given by ð(ð¥) = {Îð(ð¥)}, ð¥ â ð. Then ð¿ = ðð: ðð â
ðð is a bounded linear operator such that ð¿({Îð(ð¥)}) = {Îð(ð¥)}, ð¥ â ð.
Conversely, by hypothesis {Îð(ð¥)} â ðð, ð¥ â ð. Also, for each ð¥ â ð, we
have
(âðââ1 â ðŸ)âð¥âð = âðââ1âð¥âð â ðŸâð¥âð
†â{Îð(ð¥)}âððâ â{(Îð + Îð)(ð¥)}âðð
†â{Îð(ð¥)}âðð
†â{(Îð + Îð)(ð¥)}âðð+ â{Îð(ð¥)}âðð
†âðââð¥âð + ðŸâð¥âð
= (âðâ + ðŸ)âð¥âð,
where ð is a coefficient mapping.
Now, put ð = ðð¿ . Then ð: ðð â ð is a bounded linear operator such that
ð({Îð(ð¥)}) = ð¥, ð¥ â ð. Hence ({Îð}, ð) is a ð âBanach frame for ð with respect
to ðð and frame bounds (âðââ1 â ðŸ) and (âðâ + ðŸ).
Next, we give a necessary and sufficient condition for a ð âBanach Bessel sequence
to be a ð âBanach frame.
Theorem 4.6 Let ({Îð}, ð) ({Îð} â ð (ð, ð»), ð: ðð â ð) be a ð âBanach frame
for ð with respect to ðð. Let {Îð} â ð (ð, ð») be a ð âBanach Bessel sequence for
ð with respect to ðð . Let ð: ðð â ðð be a bounded linear operator such that
ð({Îð(ð¥)}) = {Îð(ð¥)}, ð¥ â ð. Then, there exists a reconstruction operator ð: ðð â
ð such that ({Îð}, ð) is a ð âBanach frame for ð with respect to ðð if and only if
there exists a constant ð > 0 such that
â{(Îð â Îð)(ð¥)}âðð†ðâ{Îð(ð¥)}âðð
, ð¥ â ð.
Proof. Let ðŸ > 0 be a constant such that
â{Îð(ð¥)}âðð†ðŸâð¥âð , ð¥ â ð.
Let ðŽÎ be a lower bound for the ð âBanach frame for ({Îð}, ð). Then
ðŽÎâð¥âð †â{Îð(ð¥)}âðð
†â{Îð(ð¥)}âðð+ â{(Îð â Îð)(ð¥)}âðð
†(1 + ð)â{Îð(ð¥)}âðð.
Put ð = ðð, then ð: ðð â ð is a bounded linear operator such that ð({Îð(ð¥)}) =
5140 Ghanshyam Singh Rathore and Tripti Mittal
ð¥, ð¥ â ð. Hence ({Îð}, ð) is a ð âBanach frame for ð with respect to ðð.
Conversely, let ðŽÎ, ðµÎ; ðŽÎ, ðµÎ be bounds for the ð â Banach frames
({Îð}, ð) and ({Îð}, ð), respectively. Then
ðŽÎâð¥âð †â{Îð(ð¥)}âðð†ðµÎâð¥âð , ð¥ â ð
and
ðŽÎâð¥âð †â{Îð(ð¥)}âðð†ðµÎâð¥âð , ð¥ â ð.
Therefore
â{(Îð â Îð)(ð¥)}âðð†â{Îð(ð¥)}âðð
+ â{Îð(ð¥)}âðð
†(ðµÎ + ðµÎ)âð¥âð
†ðâ{Îð(ð¥)}âðð, ð¥ â ð,
where ð =1
ðŽÎ(ðµÎ + ðµÎ).
5. STABILITY OF ð® âBANACH BESSEL SEQUENCES
In the following result, we show that ð âBanach Bessel sequences are stable under
small perturbation of given ð âBanach Bessel sequence.
Theorem 5.1. Let {Îð} â ð (ð, ð») be a ð âBanach Bessel sequence for ð with
respect to ðð and let {Îð} â ð (ð, ð») be such that {Îð(ð¥)} â ðð, ð¥ â ð . Then
{Îð} is a ð âBanach Bessel sequence for ð with respect to ðð, if there exist non-
negative constants ð, ð (ð < 1) and ð¿ such that
â{(Îð â Îð)(ð¥)}âðð†ðâ{Îð(ð¥)}âðð
+ ðâ{Îð(ð¥)}âðð+ ð¿âð¥âð , ð¥ â ð.
Proof. Let ð be the Bessel bound for the ð âBanach Bessel sequence {Îð}. Then
â{Îð(ð¥)}âðð†â{Îð(ð¥)}âðð
+ â{(Îð â Îð)(ð¥)}âðð
= (1 + ð)ðâð¥âð + ðâ{Îð(ð¥)}âðð+ ð¿âð¥âð , ð¥ â ð
†((1 + ð)ð + ð¿
1 â ð) âð¥âð , ð¥ â ð.
Hence {Îð} â ð (ð, ð») is a ð âBanach Bessel sequence for ð with respect to ðð.
REFERENCES
On G-Banach Bessel Sequences in Banach Spaces 5141
[1] M.R. Abdollahpour, M.H. Faroughi and A. Rahimi: PG Frames in Banach
Spaces, Methods of Functional Analysis and Topology, 13(3)(2007), 201-210.
[2] Huai-Xin Cao, Lan Li, Qing-Jiang Chen and Guo-Xing Li: (p, Y)-operator
frames for a Banach space, J. Math. Anal. Appl., 347(2)(2008), 583-591.
[3] P. G. Casazza, D. Han and D. R. Larson: Frames for Banach spaces, Contem.
Math., 247 (1999), 149-182.
[4] O. Christensen: An introduction to Frames and Riesz Bases, Birkhauser, 2003.
[5] O. Christensen and Y. C. Eldar: Oblique dual frames with shift-invariant
spaces, Appl. Compt. Harm. Anal., 17(1)(2004), 48-68.
[6] Renu Chugh, S.K. Sharma and Shashank Goel: A note on g frame
sequences, Int. J. Pure and Appl. Mathematics, 70(6)(2011), 797-805.
[7] Renu Chugh, S.K. Sharma and Shashank Goel: Block Sequences and G-
Frames, Int. J. Wavelets, Multiresolution and Information Processing,
13(2)(2015), 1550009-1550018.
[8] Renu Chugh and Shashank Goel: On sum of G-frames in Hilbert spaces,
Jordanian Journal of Mathematics and Statistics(JJMS), 5(2)(2012), 115-124.
[9] Renu Chugh and Shashank Goel: On finite sum of G-frames and near exact G-
frames, Electronic J. Mathematical Anal. and Appl., 2(2)(2014), 73-80.
[10] R.R. Coifman and G. Weiss: Extensions of Hardy Spaces and their use in
Analysis, Bull. Amer. Math. Soc., 83(4)(1977), 569-645.
[11] I.Daubechies, A. Grossman and Y. Meyer: Painless non-orthogonal
expansions, J. Math. Phys., 27(1986), 1271â1283.
[12] R.J. Duffin and A.C. Schaeffer: A class of non-harmonic Fourier series, Trans.
Amer. Math. Soc., 72(1952), 341â366.
[13] Y. C. Eldar: Sampling with arbitrary sampling and reconstruction spaces and
oblique dual frame vectors, J. Fourier Anal. Appl., 9(1)(2003), 77-96.
[14] H. G. Feichtinger and K. H. Gröchenig: A unified approach to atomic
decompositions via integrable group representations, Function spaces and
Applications, Lecture Notes in Mathematics, Vol. 1302 (1988), 52-73.
[15] M. Fornasier: Quasi-orthogonal decompositions of structured frames, J. Math.
Anal. Appl., 289(2004), 180â199.
[16] K.H. Gröchenig, Describing functions: Atomic decompositions versus frames,
Monatsh. fur Mathematik, 112(1991), 1-41.
[17] P.K. Jain, S. K. Kaushik and L. K. Vashisht: On Banach frames, Indian J. Pure
Appl. Math., 37(5)(2006), 265-272.
[18] P.K. Jain, S. K. Kaushik and L. K. Vashisht: On perturbation of Banach frames,
Int. J. Wavelets, Multiresolution and Information Processing, 4(3)(2006), 559-
565.
5142 Ghanshyam Singh Rathore and Tripti Mittal
[19] P.K. Jain, S. K. Kaushik and L. K. Vashisht: Bessel sequences and Banach
frames in Banach spaces, Bulletin of the Calcutta Mathematical Society,
98(1)(2006), 87-92.
[20] P.K. Jain, S. K. Kaushik and L. K. Vashisht: On stability of Banach frames,
Bull. Korean Math. Soc., 44(1) (2007), 73-81.
[21] S. Li and H. Ogawa: Pseudo-duals of frames with applications, Appl. Comp.
Harm. Anal., 11(2001), 289-304.
[22] A.Najati, M.H. Faroughi and A. Rahimi: G -frames and stability of G -frames
in Hilbert spaces, Methods of Functional Analysis and Topology, 14(3)(2008),
271â286.
[23] G.S. Rathore and Tripti Mittal: On G-Banach frames, Poincare Journal of
Analysis and Applications(submitted).
[24] W. Sun: G-frame and g-Riesz base, J. Math. Anal. Appl., 322(1)(2006), 437â
452.
[25] W. Sun: Stability of g frames, J. Math. Anal. Appl., 326(2)(2007), 858â868.