Invariance of Fréchet Frames under Perturbation

0.14 MB·English

by  Asghar Rahimi

INVARIANCE OF FRE´CHET FRAMES UNDER
PERTURBATION
2
ASGHARRAHIMI
1
0
2
Abstract. Motivating the perturbations of frames in Hilbert and Ba-

n nachspaces,inthispaperweintroducetheinvarianceofFr´echetframes
a under perturbation. Also we show that for any Fr´echet spaces, there is
J
a Fr´echet frame and any element hasa series expansion.

8
2

]
A
F
. 1. Introduction
h
t Historically, theory of frames appeared in the paper of Duffin and Scha-
a
m effer in 1952. Around 1986, Daubechies, Grossmann, Meyer and others
[ reconsidered ideas of Duffin and Schaeffer [8], and started to develop the

wavelet and frame theory. Frames for Banach spaces were introduced by K.
1
Gro¨chenig [9] and subsequently many mathematicians have contributed to
v
3 this theory [1, 3, 4, 16]. There are some complete spaces which are not Ba-
9 nach spaces ( like Fr´echet spaces). This was the main motivation for frames
9
on Fr´echet spaces. The concept of Fr´echet frames investigated by Pilipovi´c
5
. and Stoeva in [14, 15]. Like Hilbert and Banach spaces, we will show that
1
for any Fr´echet space we can find a Fr´echet frame and every element in a
0
2 Fr´echet space has a series expansion [3].
1 In this manuscript we are interested in the problem of finding conditions
:
v under which the perturbation of a Fr´echet frame is also a Fr´echet frame.
i
X The following result [2], is one of the most general and also typical results
aboutframeperturbationsforthewholespaceHwhichgeneralizes themain
r
a results in [6, 7].
Theorem 1.1. [2] Let F := {f } be a frame for H with bounds A and B,
i i I
and G := {g } is a sequence in∈H. Suppose that there exist non-negative
i i I
∈
λ ,λ and µ with λ <1 such that
1 2 2
kXci(fi−gi)k ≤ λ1kXcifik+λ2kXcigik+µkck
i i i
2000 Mathematics Subject Classification. 42C15, 46A45.
Key words and phrases. Frame, Banach frame, Fr´echet frames, Fr´echet spaces,
Perturbation.
1
2 ASGHARRAHIMI
for each finitly supported c ∈ ℓ2(N) and λ + µ < 1. Then G is a frame
1 √A
for H with bounds
λ +λ + µ λ +λ + µ
A(1− 1 2 √A)2 and B(1+ 1 2 √B)2.
1+λ 1−λ
2 2
2. Fr´echet frames
Throughout this paper, (X,k.k) is a Banach space and (X ,k.k ) is its
∗ ∗
dual, (Θ,k|.k|) is a Banach sequence space and (Θ ,k|.k| ) is the dual of Θ.
∗ ∗
Definition 2.1. A sequence space X is called a Banach space of scalar
d
valued sequences or briefly a BK-space, if it is a Banach space and the coor-
(n)
dinate functionals are continuous on X , i.e., the relations x = {α },x =
d n j
(n)
{α }∈ X ,limx = x imply limα = α .
j d n j j
Definition 2.2. Let X be a Banach space and X be a BK-space. A
d
countable family {g } in the dual X is called an X -frame for X if
i i I ∗ d
∈
(1) {g (f)} ∈ X ,∀f ∈ X;
i ∞i=1 d
(2) the norms kfk and k{g (f)} k are equivalent, i.e., there exist
X i ∞i=1 Xd
constants A,B > 0 such that
Akfk ≤ k{g (f)} k ≤ Bkfk , ∀f ∈ X
X i ∞i=1 Xd X
AandB arecalledX -framebounds. Ifatleast(1)andtheuppercondition
d
in (2) are satisfied, {g } is called an X -Bessel sequence for X.
i ∞i=1 d
If X is a Hilbert space and X = ℓ2, (2) means that {g } is a frame,
d i ∞i=1
and in this case it is well known that there exists a sequence {f } in X
i ∞i=1
such that f = P∞i=1hf,fiigi = Phf,giifi.
Similar reconstruction formulas are not always available in the Banach
space setting. This is the reason behind the following definition:
Definition 2.3. Let X be a Banach space and X a sequence space. Given
d
a bounded linear operator S : X → X, and an X -frame {g } ⊂ X , we
d d i ∗
say that ({g } ,S) is a Banach frame for X with respect to X if
i ∞i=1 d
(2.1) S({g (f)} = f, ∀f ∈X.
i
Note that (2.1) can be considered as some kind of generalized reconstruc-
tion formula, in the sense that it tells how to come back to f ∈ X based on
the coefficients {g (f)} .
i ∞i=1
There is a relationship between these definitions, a Banach frame is an
atomic decomposition if and only if the unit vectors form a basis for the
space X . The following Proposition states this result.
d
Proposition 2.4. [4] Let X be a Banach space and X be a BK-space. Let
d
{y } ⊆ X and S : X → X be given. Let {e } be the unit vectors in
i ∞i=1 ∗ d i ∞i=1
X . Then the following are equivalent:
d
INVARIANCE OF FRE´CHET FRAMES UNDER PERTURBATION 3
(1) ({y} } ,S) is a Banach frame for X with respect to X and {e }
i ∞i=1 d i ∞i=1
is a Schauder basis for X .
d
(2) ({y } ,{S(e )} ) is an atomic decomposition for X with respect
i ∞i=1 i ∞i=1
to X .
d
It is known [4] that every separable Banach space has a Banach frame.
Theorem 2.5. Every separable Banach space has a Banach frame with
bounds A = B = 1.
The main motivation of Fr´echet frames comes from some sequences {g }
i
which are not Bessel sequences but they give rise to series expansions. For
Banach space X, let {g } ⊆ X be given and let there exist {f } ⊆ X
i ∞i=1 ∗ i ∞i=1
such that the following series expansion in X holds
∞
(2.2) f = Xgi(f)fi, ∀f ∈ X.
i=1
Validity of (2.2) does not imply that {g } is a Banach frame for X with
i ∞i=1
respect to the given sequence space. As one can see in the following exam-
ples.
Example 2.6. Let {e } be an orthonormal basis for the Hilbert space H.
i ∞i=1
Consider the sequence {g } = {e ,e ,2e ,3e ,4e ,...}. This sequence is
i ∞i=1 1 1 2 3 4
not a Banach frame for H with respect to ℓ2. However, the series expansion
in H in the form (2.2) is {f } = {e ,0, 1e , 1e ,1e ,...}.
i ∞i=1 1 2 2 3 3 4 4
Validity of(2.2)implies that{g } isaBanach frameforX withrespect
i ∞i=1
to the sequence space (cid:8){ci}∞i=1 : P∞i=1cifi converges(cid:9).
Recall, a complete locally convex space which has a countable fundamen-
tal system of seminorms is called a Fr´echet space.
Let {Ys,k.ks}s N be a sequence of separable Banach spaces such that
∈
(2.3) {0} =6 ∩s NYs ⊆ ... ⊆ Y2 ⊆ Y1 ⊆ Y0
∈
(2.4) k .k ≤k. k ≤k. k ≤ ...
0 1 2
(2.5) YF := ∩s NYs is dense in Ys ∀s∈ N.
∈
Then Y is a Fr´echet space with the sequence of norms k.k , s ∈N. We use
F s
the above sequences in two cases: Ys = Xs and Ys = Θs. Let {Xs,k.ks}s N
∈
and {Θs,k|.k|s}s N be sequences of Banach and Banach sequence spaces,
which satisfy (2.∈3)-(2.5). For fixed s ∈ N, an operator V : Θ → X
F F
is s-bounded if there exist constants K > 0 such that kV{c } k ≤
s i ∞i=1 s
K k|{c } |k for all {c } ∈ Θ . If V is s-bounded for every s ∈ N,
s i ∞i=1 s i ∞i=1 F
then V is called F-bounded. Note that an F-bounded operator is contin-
uous but the converse dose not hold in general. The book of R. Meise, D.
Vogt is a very useful text book about Fr´echet spaces [12].
TheBanach sequencespaceΘiscalledsolid ifthecondition {c } ∈ Θand
i ∞i=1
|d | ≤ |c |foralli∈ N,implythat{d } ∈Θandk|{d } k| ≤ k|{c }k| .
i i i ∞i=1 i ∞i=1 Θ i Θ
4 ASGHARRAHIMI
A BK-space which contains all the canonical vectors e and for which there
i
exists a constant λ ≥ 1 such that
k|{c }n k| ≤ λk|{c } k| , ∀n ∈ N,∀{c } ∈ Θ
i i=1 Θ i ∞i=1 Θ i ∞i=1
will be called λ-BK-space. A BK-space is called a CB-space if the set of the
canonical vectors forms a basis.
Definition 2.7. Let {Xs,k.ks}s N be a family of Banach spaces, satisfying
∈
(2.3)-(2.5) andlet {Θs,k|.|ks}s N bea family of BK-spaces, satisfying(2.3)-
∈
(2.5). A sequence {g } ⊆ X is called a pre-Fr´echet frame ( a pre-F-
i ∞i=1 F∗
frame) for X with respect to Θ if for every s ∈ N there exist constants
F F
0< A ≤ B < ∞ such that
s s
(2.6) {g (f)} ∈ Θ ,
i ∞i=1 F
(2.7) A kfk ≤ k|{g (f)} |k ≤ B kfk ,
s s i ∞i=1 s s s
for all f ∈X . The constants A and B are called lower and upper bounds
F s s
for {g } . The pre-F-frame is called tight if A = B for all s ∈ N.
i ∞i=1 s s
Moreover, if there exists a F-bounded operator S : Θ → X so that
F F
S({g (f)} )= f forallf ∈ X ,thenapre-F-frame{g }iscalled aFr´echet
i ∞i=1 F i
frame ( or F-frame ) for X with respect to Θ and S is called an F-frame
F F
operator of {g } . When (2.6) and at least the upper inequality in (2.7)
i ∞i=1
hold, then {g } is called a F-Bessel sequence for X with respect to Θ .
i ∞i=1 F F
SinceX isdenseinX foralls∈ N,g hasauniquecontinuousextension
F s i
on X , we show it by gs. Thus gs ∈ X and gs = g on X .
s i i s∗ i i F
The following Theorem gives some conditions, under which an element
can be expanded by some elementary vectors.
Theorem 2.8. [14, 15] Let {Xs,k.ks}s N be a family of Banach spaces,
∈
satisfying (2.3)-(2.5) and let {Θs,k|.|ks}s N be a family of CB-spaces, satis-
fying (2.3)-(2.5) and we assume that Θ i∈s a CB-space for every s∈ N. Let
∗s
{g } be a pre-F-frame for X with respect to Θ . There exists a family
i ∞i=1 F F
{f } ⊂ X such that
i ∞i=1 F
(1) f = P∞i=1gi(f)fi and g =P∞i=1g(fi)gi, ∀f ∈ XF and ∀g ∈ XF∗;
(2) f = P∞i=1gis(f)fi and g = P∞i=1g(fi)gis, ∀f ∈ Xs and ∀g ∈ Xs∗,∀s ∈
N;
(3) for every s ∈ N, {f } is a Θ -frame for X .
i ∞i=1 ∗s s∗
if and only if there exists a continuous projection U from Θ onto its sub-
F
space {g (f)} :f ∈ X .
(cid:8) i ∞i=1 F(cid:9)
The following proposition shows that the pre-Fr´echet Besselness is equiv-
alent to the F-boundedness of an operator.
Proposition 2.9. Let{Xs,k.ks}s N be afamily of Banach spaces, satisfying
∈
(2.3)-(2.5) and let {Θs,k|.|ks}s N be a family of CB-spaces, satisfying (2.3)-
(2.5) and we assume that Θ ∈is a CB-space for every s ∈ N. The family
∗s
{g } ⊆ X is a pre-Fr´echet Bessel sequences for X with respect to Θ
i ∞i=1 F∗ F F
INVARIANCE OF FRE´CHET FRAMES UNDER PERTURBATION 5
if and only if the operator T :Θ∗F → XF∗ defined by T{di}∞i=1 = P∞i=1digi is
well defined and kTk ≤ B , for all s ∈ N.
s s
Proof. First, suppose that {g } ⊆ X is a pre-Fr´echet Bessel sequences
i ∞i=1 F∗
for X with respect to Θ . Define the operator
F F
R :X → Θ
F F
by
Rf = {g (f)} .
i ∞i=1
Since {g (f)} is a pre-Fr´echet Bessel sequence , so kRk ≤ B for every
i ∞i=1 s s
s∈ N. The adjoint of R is in the form R :Θ → X and
∗ ∗F F∗
R (e )f = e (Rf)=e ({g f} )= g f
∗ j j j i ∞i=1 j
, so R e = g . Put T = R , then kTk ≤ B and
∗ j j ∗ s s
∞ ∞ ∞ ∞
T{di}∞i=1 = T(Xdiei)= XdiTei = XdiR∗ei = Xdigi.
i=1 i=1 i=1 i=1
Conversely, suppose the operator T : Θ → X defined by T{d } =
∗F F∗ i ∞i=1
P∞i=1digi is well defined and s-bounded for all s ∈ N . It is clear that
Te = g and T : X → Θ is (T f)e = f(Te ) = f(g ). Therefore
i i ∗ F∗∗ ∗F∗ ∗ i i i
{g (f)} = {(T f)e } , i.e. {g (f)} ∈ Θ . The s- boundedness of T
i ∞i=1 ∗ i ∞i=1 i ∞i=1 F
imply that k|{g (f)} |k ≤ B . (cid:3)
i ∞i=1 s s
SimilartoTheorem2.5,thefollowingTheoremshowsthatforanyFr´echet
space X we can construct a Fr´echet frame.
F
Theorem 2.10. Let {Xs,k.ks}s N be a family of separable Banach spaces
∈
satisfying (2.3)-(2.5). Let X := X . Then X can be equipped with
F Ts N s F
a Fr´echet frame with respect to an ap∈propriately sequence space.
Proof. Since X is a separable Banach space for any s ∈ N, there is a se-
s
quence {xs} ⊆ X , such that {xs} = X . For any f ∈ X there is
i ∞i=1 s i ∞i=1 s s s
a subsequence xs → f as i → ∞. By Hahn-Banach Theorem, there is
ki s
gs ∈ X such that gs(xs)= kxsk and kgsk = 1. Now,
i s∗ i i i i
kxs k= |gs (x )|≤ kgs (fs)k+kfs−xs k
ki ki ki ki ki
, so kfsk ≤ supkg (fs)k. Since we also have kfsk ≥ sup kg (fs)k, therefore
i i i
kfsk = supkgs(fs)k, ∀fs ∈ X .
i s
i
Let Θs ⊆ ℓ∞ and Θs = (cid:8){gi(fs)} : fs ∈ Xs(cid:9), then {Θs}s NΘs satisfies
(2.3)-(2.5). Let Θ := and S({g (f)}) = f. Then ∈({gs},S) is a
F Ts N i i
Fr´echet frame for X with r∈espect Θ . (cid:3)
F F
6 ASGHARRAHIMI
3. Perturbation of Fr´echet Frames
Like the results about p-frames [18], Banach frames and atomic decom-
positions [5, 10], generalized frames [13], continuous frames [17], we study
perturbations of Fr´echet frames. We need the following assertion.
Lemma 3.1. [11] Let U : X → X be a linear operator and assume that
there exist constants λ ,λ ∈ [0,1[ such that
1 2
kx−Uxk ≤ λ kxk+λ kUxk, ∀x ∈ X.
1 2
Then U is invertible and
1−λ 1+λ
1 1
kxk ≤ kUxk ≤ kxk
1+λ 1−λ
2 2
1−λ 1+λ
2kxk ≤ kU 1xk ≤ 2kxk
−
1+λ 1−λ
1 1
for all x ∈ X.
The simplest assertion about perturbation of F-Bessel sequences is:
Proposition 3.2. Let X be a Fr´echet space satisfying (1)-(3) and let Θ
F F
be a Fr´echet sequence space satisfying (1)-(3) so that Θ is a reflexive CB-
s
space for every s. Let {g } be an Fr´echet Bessel sequence for X with respect
i F
to Θ with bounds B and let {f }⊂ X .
F s i F∗
Assume that {(g −f )(f)} ∈ Θ , ∀f ∈ X , and ∃µ ≥0 such that
i i F F s
e
(3.1) k|{(g −f )(f)}|k ≤ µ kfk , ∀f ∈ X
i i s s s F
e
(i.e. {g −f } is an Fr´echet Bessel sequence for X w.r.t. Θ ). Then {f }
i i F F i
is an F-Bessel sequence for X w.r.t. Θ with bounds B +µ .
F F s s
The converse also holds. e
Proof. It is clear that {(f )(f)} = {(−g +f )(f)}+{(g −f )(f)} ∈ Θ for
i i i i i F
all f ∈ X , also
F
|k{f (f)}|k ≤ |k{(g −f )(f)}|k +|k{(g −f )(f)}|k ≤ (B +µ )kfk
i s i i s i i s s s
for all s ∈ N,f ∈ X . e (cid:3)
F
The following Theorem generalizes a Theorem of [10] to Fr´echet frames
and gives a necessary and sufficient condition for the stability of Fr´echet
frames.
Theorem 3.3. Let {g },S be a Fr´echet frame for X with respect to Θ
(cid:0) i (cid:1) F F
with bounds A and B . Let {h } ⊆ X such that {h (f)} ∈ Θ for all
s s i F∗ i F
f ∈ X and let D : Θ → Θ be a continuous linear operator such that
F F F
D{h (f)} = {g (f)}, f ∈ X . Then there exists an operator V : Θ → X
n n F F F
such that ({h },V) is a Fr´echet frame if and only if for each s ∈ N there
n
exists λ > 0 such that
s
(3.2) k|{(g −h )(f)}|k ≤ λ min{k|{g (f)}|k ,k|{(h (f))}|k }
n n s s n s n s
for all f ∈ X .
F
INVARIANCE OF FRE´CHET FRAMES UNDER PERTURBATION 7
Proof. Suppose there exists λ for every s ∈ N such that (3.2) holds. By
s
assumption, {h (f)}∈ Θ for all f ∈ X . For any f ∈ X and s∈ N,
i F F F
A kfk ≤ k|{g (f)}|k
s s n s
≤ k|(g −h )(f)}|k +k|{h (f)}|k
n n s n s
≤ λ k|{h (f)}|k +k|{h (f)}|k
s n s n s
= (1+λ )k|{h (f)}|k
s n s
≤ (1+λ ) k|{(g −h )(f)}|k +k|{g (f)}|k
s (cid:0) n n s n s(cid:1)
≤ (1+λ )2k|{g (f)}|k
s n s
≤ (1+λ )2B kfk .
s s s
So we have
A
s
kfk ≤ k|{h (f)}|k ≤ (1+λ )B kfk .
s n s s s s
1+λ
s
Let V = SD. Then V({h (f)}) = SD({h (f)}) = f, i.e. {h },V is a
n n (cid:0) n (cid:1)
Fr´echet frame for X with respect to Θ .
F F
Conversely, suppose {g },S and {h },V are Fr´echet frames for X with
(cid:0) i (cid:1) (cid:0) n (cid:1) F
respect to Θ with bounds A ,B and A ,B , respectively. Then, by using
F s s ′s s′
the inequalities, we get
B
k|{(g −h )(f)}|k ≤ 1+ s′ k|{g (f)}|k , f ∈ X
n n s (cid:0) (cid:1) n s F
A
s
and
B
s
k|{(g −h )(f)}|k ≤ 1+ k|{h (f)}|k , f ∈ X .
n n s (cid:0) (cid:1) n s F
A
′s
Chooseλ := Max{1+Bs′,1+Bs},therefore(3.2)holdsforanyf ∈ X . (cid:3)
s As A′s F
Theorem 3.4. Let {g },S be a Fr´echet frames for X with respect to Θ .
(cid:0) i (cid:1) s s
Let {h } ∈ X such that {h (f)} ∈ Θ for all f ∈ X and let D : Θ →
i F∗ i F F F
Θ be a continuous ( or F-bounded) linear operator such that D{g (f)} =
F n
{h (f)}, f ∈ X . Let {α } and {β } be sequences of positive numbers for
n F n n
which 0 < infα ≤ supα < ∞ and 0 < infβ ≤ supβ < ∞. If for any
n n n n
s∈ N there exist non negative scalers λ ,µ ∈ [0,1[ and γ such that
s s s
(1) kSk γ < (1−λ )(infα )
s s s n
(2) k|{(α g − β h )f}|k ≤ λ k|{(α g )f}|k + µ k|{(β h )f}|k +
n n n n s s n n s s n n s
γ kfk ,∀f ∈ X ,
s s F
then there is a F-bounded operator V : Θ → X such that ({h },V) is a
F F i
Fr´echet frame for X with respect to Θ .
F F
Proof. Let W : X → Θ with Wf := {g (f)} for all f ∈ X . Then
F F i F
SW : Θ → Θ is an identity operator and for all s∈ N
F F
kfk =kSWfk ≤ kSk k|{g (f)}|k .
s s s i s
8 ASGHARRAHIMI
Now,
k|{(β h )f}|k ≤ k|{(α g )f}|k +k|{(α g −β h )f}|k
n n s n n s n n n n s
≤ k|{(α g )f}|k +λ k|{(α g )f}|k
n n s s n n s
+ µ k|{(β h )f}|k +γ kfk
s n n s s s
for all s ∈ N and f ∈ X . Therefore
F
(1−µ )k|{(β h )f}|k ≤ (1+λ )kWk (supα )+γ kfk ,
s n n s (cid:0) s s n s(cid:1) s
also
(1−µ ) infβ k|{(h )f}|k ≤ (1−µ )k|{(β h )f}|k
s (cid:0) n(cid:1) n s s n n s
≤ (1+λ )kWk (supα )+γ kfk ,
(cid:0) s s n s(cid:1) s
By using (2), we have:
(1+µ )k|{(β h )f}|k ≥ (1−λ )k|{(α g )f}|k −γ kfk
s n n s s n n s s s
≥ (1−λ )kSk 1(infα )−γ kfk
(cid:0) s −s n s(cid:1) s
for all f ∈X and s ∈ N. Therefore
F
(1+µ ) supβ k|{(h )f}|k ≥ (1+µ )k|{(β h )f}|k
s (cid:0) n(cid:1) n s s n n s
≥ (1−λ )kSk 1 infα −γ kfk
(cid:0) s − (cid:0) n(cid:1) s(cid:1) s
for all f ∈ X and s ∈ N. The above inequality shows the frame bounds.
F
Let V = SD. Then V is a bounded operator that V{h (f)} = f, for all
n
f ∈ X . Therefore ({h },V) is a Fr´echet frame for X with respect to
F i F
Θ . (cid:3)
F
Theorem 3.5. Let ({g },V) be a Fr´echet frame for X with respect to Θ .
i F F
Suppose λ ,λ ,µ ≥ 0 such that max{λ ,λ +µ B } < 1 for all s ∈ N
1s 2s s 2s 1s s s
and S : Θ → X a continuous operator such that for any {c } ∈ Θ and
F F i F
s∈ N
(3.3) kS{c }−V{c }k ≤ λ kV{c }k +λ kS{c }k +µ k|{c }|k
i i s 1 i s 2s i s s i s
then there exists a {h } ⊆ X such that ({h },S) is a Fr´echet frame of X
i F∗ i F
with respect to Θ .
F
Proof. For f ∈ X , let c = g (f) in (3.3), then we have
F i i
kS{g (f)}−V{g (f)}k ≤λ kV{g (f)}k +λ kS{g (f)}k +µ k|{g (f)}|k
i i s 1s i s 2s i s s i s
since V({g (f)}) = f, so
i
kS{g (f)}−fk ≤ λ kf}k +λ kS{g (f)}k +µ B kf| .
i s 1s s 2s i s s S s
Let L(f) := S{g (f)}, so
i
kf −Lfk ≤ λ kfk +λ kLfk +µ B kf|
s 1s s 2s s s S s
or
kf −Lfk ≤ (λ +µ B )kfk +λ kLfk .
s 1s s s s 2s s
INVARIANCE OF FRE´CHET FRAMES UNDER PERTURBATION 9
Lemma 3.1 results that the operator L is invertible and
1−λ 1+λ
2s kfk ≤ kL 1fk ≤ 2s kfk
s − s s
1+λ +µB 1−(λ +µ )B
1s s 1s s s
and f = LL 1f = S({g (L 1f)}). It is clear that {g (L 1f)} ⊆ X . By
− i − i − F∗
choosing h = g ◦L 1, we have
i i −
A (1−λ )
k|{h (f)}|k = k|{g (L 1f)}|k ≥ A k|L 1|k ≥ s 2s kfk
i s i − s s − s s
1+λ +µ B
1s s s
and
B (1+λ )
k|{h (f)}|k ≤ B kL 1fk ≤ s 2s kfk .
i s s − s s
1−(λ +µ B )
1s s s
(cid:3)
Theorem 3.6. Let ({g },S) be a F-frame for X with respect to Θ . Let
i F F
{h }⊂ X . If for every s ∈ N there exist λ ,µ ≥ 0 such that
i F∗ s s
(1) λ kUk+µ ≤ kSk 1,
s s −s
(2) k|{(g −h )f}|k ≤ λ k|{(g )f}|k +µ kfk for all s ∈N and f ∈ X .
i i s s i s s s F
Then there is a continuous operator T such that ({h },T) is a F-frame for
i
X with respect to Θ .
F F
Proof. It is a direct result of (2) that the operator V : X → Θ defined by
F F
Vf := {h (f)} is bounded and
i
k|Uf −Vf|k ≤ λ k|Uf|k +µ kfk
s s s s s
for all s ∈ N and f ∈ X . Therefore,
F
(3.4) k|Vf|k ≤ (kUk +λ kUk +µ )kfk
s s s s s s
for all s ∈ N and f ∈ X . Also, SU = I imply that
F
kI −SVk ≤ kSk (λ kUk +µ ) < 1.
s s s s s
Therefore SV is invertible and k(SV)−1ks ≤ (cid:0)1 − (λkUks + µ)kSks)(cid:1)−1.
Finally T = (SV) 1S and TV = I. For all f ∈ X and s ∈ N
− F
kSk
(3.5) kfk ≤ kTk .k|Vf|k ≤ k|Vf|k .
s s s s
1−(λ kUk +µ )kSk
s s s s
(3.4) and (3.5) which imply that for every s ∈ N and f ∈ X :
F
1−(λ kUk +µ )kSk
s s s s
kfk ≤ k|{h (f)}|k ≤ (kUk +λ kUk +µ )kfk .
s i s s s s s s
kSk
s
So ({h },T) is a F-frame for X with respect to Θ .
i F F
(cid:3)
Acknowledgment: Some of the results in this paper were obtained dur-
ingtheauthor’svisitingattheAcoustics Research Institute, AustrianAcad-
emy of Sciences, Austria ( Summers 2009, 2010 and 2011). He thanks this
institute for hospitality. Also, he would like to thank Peter Balazs and
D. T. Stoeva for useful discussions, comments and suggestions. This work
10 ASGHARRAHIMI
was partly supported by the WWTF project MULAC (’Frame Multipliers:
Theory and Application in Acoustics; MA07-025)
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Department of mathematics ,University of Maragheh, Maragheh, Iran
E-mail address: [email protected]