INVARIANCE OF FREĆHET FRAMES UNDER PERTURBATION 2 ASGHARRAHIMI 1 0 2 Abstract. Motivating the perturbations of frames in Hilbert and Ba- n nachspaces,inthispaperweintroducetheinvarianceofFréchetframes a under perturbation. Also we show that for any Fréchet spaces, there is J a Fréchet 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 Fréchet spaces). This was the main motivation for frames 9 on Fréchet spaces. The concept of Fréchet frames investigated by Pilipović 5 . and Stoeva in [14, 15]. Like Hilbert and Banach spaces, we will show that 1 for any Fréchet space we can find a Fréchet frame and every element in a 0 2 Fréchet 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 Fréchet frame is also a Fréchet 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, Fréchet frames, Fréchet 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. Fréchet 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 reconstruction 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 FREĆHET 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 Fréchet 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 Fréchet 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 Fréchet 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 continuous but the converse dose not hold in general. The book of R. Meise, D. Vogt is a very useful text book about Fréchet 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-Fréchet 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 aFréchet 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, satisfying (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-Fréchet Besselness is equivalent 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-Fréchet Bessel sequences for X with respect to Θ i ∞i=1 F∗ F F INVARIANCE OF FREĆHET 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-Fréchet 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-Fréchet 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,thefollowingTheoremshowsthatforanyFréchet space X we can construct a Fréchet 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 Fréchet 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 Fréchet frame for X with r∈espect Θ . (cid:3) F F 6 ASGHARRAHIMI 3. Perturbation of Fréchet Frames Like the results about p-frames [18], Banach frames and atomic decompositions [5, 10], generalized frames [13], continuous frames [17], we study perturbations of Fréchet 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 Fréchet space satisfying (1)-(3) and let Θ F F be a Fréchet sequence space satisfying (1)-(3) so that Θ is a reflexive CB- s space for every s. Let {g } be an Fréchet 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 Fréchet 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 Fréchet frames and gives a necessary and sufficient condition for the stability of Fréchet frames. Theorem 3.3. Let {g },S be a Fréchet 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 Fréchet 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 FREĆHET 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) Fréchet frame for X with respect to Θ . F F Conversely, suppose {g },S and {h },V are Fréchet 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 Fréchet 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 Fréchet 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 Fréchet frame for X with respect to F i F Θ . (cid:3) F Theorem 3.5. Let ({g },V) be a Fréchet 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 Fréchet 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 FREĆHET 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) References [1] A. Aldroubi, Q. Sun and W. Tang, p-frames and shift invariant subspaces of Lp, J. Fourier Anal. Appl.7 no. 1, (2001) 1-22. [2] P.G.CasazzaandO.Christensen,Perturbation ofoperators andapplicationtoframe theory, J. Fourier Anal. Appl.3(5) (1997) 543–557. [3] P. G. Casazza, O. Christensen and D. T. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal. Appl.307 (2005) 710723. [4] P. G. Casazza, D. Han and D. Larson, Frames for Banach spaces, Contemp. Math. 247 (1999) 149182. [5] O. Christensen, An Introduction to Frame and Riesz Bases, Birkhauser 2002. [6] O. Christensen, Frame perturbations, Proc. Amer. Math. Soc. 123 (1995) 12171220. [7] O. Christensen, A PaleyWiener theorem for frames, Proc. Amer. Math. Soc. 123 (1995) 21992201. [8] R.J.DuffinandA.C.Schaeffer,AclassofnonharmonicFourierseries,Trans.Amer. Math. 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Anal.Top. Vol.12, No. 2 (2006) 170-182. [18] D. T. Stoeva, Perturbation of frames in Banach spaces, arXiv:0902.3602 (2009). Department of mathematics ,University of Maragheh, Maragheh, Iran E-mail address: asgharrahimi@yahoo.com

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