g-frame Sequence Operators, cg-riesz Bases and Sum of cg-frames

Size: px

Start display at page:

Download "g-frame Sequence Operators, cg-riesz Bases and Sum of cg-frames"

Erick Blake
5 years ago
Views:

1 International Mathematical Forum, Vol. 6, 2011, no. 68, g-frame Sequence Operators, cg-riesz Bases and Sum of cg-frames M. Madadian Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran M. Rahmani Department of Mathematics, University of Tabriz Tabriz, Iran m rahmani@tabrizu.ac.ir Abstract In this paper we study the operators associated with g-frame sequences in a Hilbert space H, i.e., the synthesis operator, the analysis operator and the g-frame operator. Also, for all of these operators, we investigate their pseudo-inverses. Furthermore, we introduce concept of cg-riesz bases and verify some properties of them. Finally, for constructing a large number of cg-frames from existing cgframes, we give necessary and sufficient conditions on cg-bessel families {Λ ω B(H, H ω ) : ω } and {θ ω B(H, H ω ) : ω } and operators L 1 and L 2 on H such that {Λ ω L 1 + θ ω L 2 } ω is a cg-frame for H. Mathematics Subject Classification: 42C15, 41A58, 47A05 Keywords: g-frame sequence, pseudo-inverse, cg-frame, cg-riesz basis 1 Introduction The concept of frames (discrete frames) in Hilbert spaces has been introduced by Duffin and Schaeffer [5] in 1952 to study some deep problems in nonharmonic Fourier series. After the fundamental paper [4] by Daubechies, Grossman and Meyer, frame theory began to widely used. Sun introduced a g-frame and a g-riesz basis in a complex Hilbert space and discussed some properties of them ([9]). Also, continuous g-frames are introduced in [1]. In this paper

2 3358 M. Madadian and M. Rahmani we generalize some results in [2] and [7] to g-frame sequences and cg-frames and we introduce concept of cg-riesz bases. Throughout this paper, H is a Hilbert space, (, µ) is a measure space with positive measure µ and {H i }, {H ω } ω are two families of Hilbert spaces. Definition 1.1. We call {Λ i B(H, H i ) : i I} a generalized frame, or simply a g-frame, for H with respect to {H i } if there are two positive constants A and B such that A f 2 Λ i f 2 B f 2, f H. (1.1) We call A and B the lower and upper frame bounds, respectively. We call {Λ i B(H, H i ) : i I} a λ-tight g-frame if A = B = λ and we call it a Parseval g-frame if A = B = 1. If we have only the second inequality in (1.1), we call it a g-bessel sequence. For a sequence {H i }, define ( ) H i = { {f i } f i H i, {f i } 2 2 = f i 2 < }, l 2 it is easy to show that with pointwise operations and inner product as < {f i }, {g i } >= < f i, g i >, ( H i) l 2 is a Hilbert space. Definition 1.2. We say {Λ i B(H, H i ) : i I} is a g-frame sequence in H, if it is a g-frame for span{λ i (H i )}. We define the synthesis operator for a g-bessel sequence {Λ i B(H, H i ) : i I} as follows ( T Λ : H i H )l 2 T Λ ({f i } ) = Λ i (f i ). (1.2) This series converges unconditionally in H. It is easy to show that the adjoint operator of T Λ is given by ( ) TΛ : H H i l 2

3 g-frame sequence operators, cg-riesz bases 3359 T Λ(f) = {Λ i f}. (1.3) The operator T Λ is called the analysis operator of {Λ i}. Also, the g-frame operator of {Λ i } is defined as follows: S : H H S Λ = T Λ T Λf = Λ i Λ i f, (1.4) which is a bounded, self-adjoint, positive and invertible operator and A S B. Definition 1.3. Let ϕ Π ω H ω. We say that ϕ is strongly measurable if ϕ as a mapping of to ω H ω is measurable, where Π ω H ω = { f : ω H ω ; f(ω) H ω }. Definition 1.4. We call {Λ ω B(H, H ω ) : ω } a continuous generalized frame, or simply a cg-frame, for H with respect to {H ω } ω if: (i) for each f H, {Λ ω f} ω is strongly measurable, (ii) there are two positive constants A and B such that A f 2 Λ ω f 2 dµ(ω) B f 2, f H. (1.5) We call A, B lower and upper cg-frame bounds, respectively. Define ( ω H ω, µ ) L 2 = { ϕ Π ω H ω ϕ is strongly measurable, ϕ 2 2 = ϕ(ω) 2 dµ(ω) < }. ( ω H ω, µ ) is a Hilbert space with inner product L 2 < ϕ, ψ >= < ϕ(ω), ψ(ω) > dµ(ω). Proposition 1.5. ([1]) Let {Λ ω } ω be a cg-bessel family with respect to {H ω } ω for H with bound B. Then the mapping T of ( ω H ω, µ ) to H L 2 defined by < T ϕ, h >= < Λ ωϕ(ω), h > dµ(ω), ϕ ( ω H ω, µ ), h H (1.6) L 2 is linear and bounded with T B. Furthermore for each h H and ω T (h)(ω) = Λ ω h. (1.7) The operators T and T are called synthesis and analysis operators of cg-bessel family {Λ ω } ω, respectively.

4 3360 M. Madadian and M. Rahmani Let {Λ ω } ω be a cg-frame with respect to {H ω } ω for H with frame bounds A, B. The operator S : H H defined by < Sf, g >= < f, Λ ωλ ω g > dµ(ω), f, g H (1.8) is called the cg-frame operator of {Λ ω } ω which is a positive operator and invertible. Lemma 1.6. ([3]) Let H, K be Hilbert spaces, and suppose that U : K H is a bounded operator with closed range R U. Then there exists a bounded operator U : H K for which keru = R U, R U = (keru), UU f = f, f R U. We call the operator U the pseudo-inverse of U. This operator is uniquely determined by these properties. Lemma 1.7. ([3]) Let U : K H be a bounded operator with closed range. Then the following holds: (i) The orthogonal projection of H onto R U is given by UU. (ii) The orthogonal projection of K onto R U is given by U U. (iii) U has closed range, and (U ) = (U ). (iv) On R U, the operator U is given explicitly by U = U (UU ) 1. 2 g-frame sequence operators Let {Λ i B(H, H i ) : i I} be a g-frame sequence in H, V = span{λ i (H i )}, ι V : V H the inclusion operator, the analysis operator, the synthesis operator and ι V (f) = f ( ) U : V H i l 2 U(f) = {Λ i f} ( T : H i V )l 2 T ({f i } ) = Λ i (f i ) S : V V

5 g-frame sequence operators, cg-riesz bases 3361 S(f) = Λ i Λ i (f) the g-frame operator. In the following we verify some basic relations of these operators. Proposition 2.1. If {Λ i } is a g-frame sequence in H, then (i) T = ι V T. (ii) R T = R T = V and P is the projection on V. (iii) U = UP. (iv) R U = R U. (v) S = T U = ι V T UP = ι V SP. Proof. (i), (ii) and (v) are clear. (iii) and (iv): For g V, we have U(g) = {Λ i g} = 0. Every f H, can be written uniquely as f = g + h with g V and h V, therefore U(f) = U(g) + U(h) = U(h) = UP (f). As {Λ i } is a g-frame sequence, so it is a g-frame for V and hence the operators U and T are bounded, T = U and U = T. Moreover Corollary 2.2. Let {Λ i } be a g-frame sequence. Then the following holds: (i) The operator U is bounded. (ii) The operator T is bounded. (iii) T = U and U = T. Proof. (i) and (ii): By boundedness of T, U, P and ι V and Proposition 2.1, U and T are bounded. (iii): U = (UP ) = P U = ι V T = T. Similarly U = T. According to {Λ i } is a g-frame for V, there is a sequence { Λ i } which is the canonical dual g-frame for V, Λ i = Λ i S 1. { Λ i } is again a g-frame sequence in H such that span{ Λ i (H i )} = V and bounds Ã = 1 B and B = 1 A. Let T, Ũ, S, T, Ũ and S be the corresponding operators associated with this g- frame sequence. Then T = S 1 T, Ũ = US 1, S = S 1 and T Ũ = T U = id V. Corollary 2.3. Let {Λ i } be a g-frame sequence in H and { Λ i } be its dual sequence, then the following properties hold: (i) T = ι V T and Ũ = ŨP. (ii) S = ι V S 1 P. (iii) T Ũ = ι V P = T U.

6 3362 M. Madadian and M. Rahmani It follows from Property (iii) above that the projection P onto V as a function ( from H into H is T Ũ. Also, denoting by Q the orthogonal projection of i) H onto (kert ) = R T, it is straightforward to show that Q is U T = U T. l 2 Theorem 2.4. The following statements are equivalent: (i) {Λ i } is a g-frame sequence in H with bounds A and B. (ii) There exist positive constants A and B such that A P f 2 Uf 2 B P f 2, f H. (iii) There exist positive constants A and B such that A Q{f i } 2 T {f i } 2 B Q{f i } 2, ( {f i } H i. )l 2 Proof. (i) (ii): Since U = UP, so by definition of g-frame sequence, (ii) is clear. (ii) (iii): By (ii), {Λ i } is a g-frame sequence in H, so its dual { Λ i } is a g-frame sequence and 1 B P f 2 Ũf 2 1 A P f 2, f H. ( ) Choose {f i } H i and let f = T ({f i } ). Then l 2 1 B P T {f i} 2 ŨT {f i} 2 1 A P T {f i} 2. Hence or 1 B T {f i} 2 Q{f i } 2 1 A T {f i} 2, A Q{f i } 2 T {f i } 2 B Q{f i } 2. (iii) (i): Q is bounded, so T is continuous. For {f i } (kert ) A {f i } 2 T {f i } 2 B {f i } 2. Therefore T (kert ) is bounded, injective and has closed range. By R T (kert ) = R T, T is bounded and has closed range. By Lemma (1.2) in [6], {Λ i } is a g-frame sequence in H.

7 g-frame sequence operators, cg-riesz bases 3363 Corollary 2.5. The followings are equivalent: (i) {Λ i } is a g-frame sequence in H. (ii) T is continuous, has closed rang and T 1 P f Uf T P f, f H. (iii) T is continuous, has closed rang and T 1 Q{f i } T {f i } T Q{f i }, ( {f i } H i. )l 2 Proof. (i) (ii): By (i) and Theorem 2.4, U = T has closed range, so by Lemma 1.6, we have inf Uf Uf 2 2 = inf f =1 f 0 f = inf Uf 2 0 f (keru) f F 2 = inf F 0 U F = 1 = 1 U sup F F 0 U = (T ) 1 = T 1. F 2 Also, for each f H sup Uf 2 = sup f =1 f =1 = sup sup F 2 =1 f =1 sup F 2 =1 All the other proofs are straightforward. < Uf, F > < f, U F > = sup U F = U = T. F 2 =1 For a g-frame sequence the g-frame operator S = T T is bounded and self adjoint. Moreover R S = T T (H) = T (T H + kert ) = T (R T + RT ( ) = T ( H i ) = R T )l 2 By above corollary R T is closed so S has closed range. Hence, S has a bounded pseudo-inverse S. In the following we state some properties of S. Lemma 2.6. The operator S : H H has the following properties: (i) S = S = ι V S 1 P. (ii) SS = S S = P. (iii) S (I P ) = 0. (iv) S P = P S = S.

8 3364 M. Madadian and M. Rahmani Proof. (i): Since T = ι V T and R T = V, so R T = V and R S = R S. Also, S S = ι V SP ι V SP = ιv S SP = ι V SS 1 P = ι V P. In addition ker S = {f : Sf = 0} = {f : ι V SP f = 0} = {f : P f = 0} = V. Similarly, kers = V. So by Lemma 1.6, S = S. (ii): By (i) and Lemma 1.6, SS is the orthogonal projection onto R S = R T, so SS = P. By taking adjoint we have S S = P. (iii): S (I P ) = S S P = S S SS = S S = 0. (iv): By (ii), S P = S so R S P = R S = R S. Hence by (ii), Lemma 1.6 and (i), S = S P = P S P = P S SS = P S. Proposition 2.7. Let {Λ i } be a g-frame sequence in H. Then (i) T = Ũ and U = T. (ii) T = T S. (iii) (T ) T = S. (iv) (T ) = S T. (v) T 2 = S. (vi) T 2 = S. Proof. (i): By Corollary 2.3, T Ũ = ι V P and kerũ = V. Since kert = ker T so RŨ = (kert ). So by Lemma 1.6, T = Ũ. Similarly, U = T. (ii): R S = R S = V, so P = SS = (T T )S = T (T S ). Hence by Lemma 1.7, T = T S. (iii): By (i), it is clear. (iv): By (ii), it is clear. (v): For each f H, T f 2 =< Sf, f > Sf f S f 2, so T 2 = T 2 S. Also, S = T T T 2. (vi): For each f H, T f 2 =< T S f, T S f >=< T T S f, S f > =< P f, S f >=< f, P S f >=< f, S f >, hence, T f 2 =< f, S f >, which implies T 2 S. S is self adjoint, so Therefore, T 2 = S. S = sup < S f, f > = sup T f 2 T 2. f =1 f =1

9 g-frame sequence operators, cg-riesz bases cg-riesz bases Definition 3.1. A family Λ = {Λ ω B(H, H ω ) : ω } is called a cg-riesz basis for H, if: (i) {h : Λ ω h = 0, a.e. [µ]} = {0}, (ii) for each h H, {Λ ω h} ω is strongly measurable and the operator T : ( ω H ω, µ ) L 2 H defined by (1.6), is well-defined and there are positive constants A and B such that A ϕ 2 T ϕ B ϕ 2, ϕ ( ω H ω, µ ) L 2. Lemma 3.2. Suppose (, µ) is a measure space where µ is σ-finite and consider the family {Λ ω } ω. (i) Assume that for each h H, {Λ ω h} ω is strongly measurable. {Λ ω } ω is a cg-riesz basis for H if and only if the operator T defined by (1.6) is an invertible bounded operator from ( ω H ω, µ ) L 2 onto H. (ii) If {Λ ω } ω is a cg-riesz basis for H then {Λ ω } ω is a cg-frame for H. Proof. (i) By proof of Proposition 2.12 in [1] and Proposition 1.5 and Theorem 4.12 in [8], it is clear. (ii) By assumption and (i), the operator T defined by (1.6) is a invertible bounded operator. So by Theorem 2.12 in [1], {Λ ω } ω is a cg-frame for H. Theorem 3.3. Suppose (, µ) is a measure space where µ is σ-finite. Let {Λ ω } ω be a cg-frame for H with synthesis operator T. Then the following statements are equivalent: (i) {Λ ω } ω is cg-riesz basis for H. (ii) T is one-to-one. (iii) R T = ( ω H ω, µ ). L 2 Proof. (i) (ii): It is obvious. (ii) (i): By Theorem 2.12 in [1], the operator T defined by (1.6) is bounded and onto. By (ii), T is also one-to-one. Therefore T has a bounded inverse T 1 : H ( ω H ω, µ ) and hence {Λ L 2 ω } ω is a cg-riesz basis H by Lemma 3.2. (i) (iii): By Lemma 3.2, T has a bounded inverse on R T = H. So R T ( = ω H ω, µ ). L 2 (iii) (i): Since the operator T is bijective, so T is invertible.

10 3366 M. Madadian and M. Rahmani 4 Sum of cg-frames Proposition 4.1. Let {Λ ω } ω be a cg-frame for H with cg-frame operator S and frame bounds A and B and L : H H be a bounded operator. Then {Λ ω L} ω is a cg-frame for H if and only if L is onto. Moreover, in this case the cg-frame operator of {Λ ω L} ω is L SL and its bounds are L 2 A and L 2 B. Proof. If L is onto then for each h H Λ ω Lh 2 dµ(ω) A Lh 2 ( L 2 A) h 2, also Λ ω Lh 2 dµ(ω) B Lh 2 ( L 2 B) h 2. Hence {Λ ω L} ω is a cg-frame for H with frame bounds L 2 A and L 2 B. Conversly, let {Λ ω L} ω be a cg-frame for H. Then for each h H and ϕ ( ω H ω, µ ), we have L 2 < T LΛ ϕ, h >= < L Λ ωϕ(ω), h > dµ(ω) = < Λ ωϕ(ω), Lh > dµ(ω) = < L T Λ ϕ, h >. So T LΛ = LT Λ. Since {Λ ω L} ω is a cg-frame so T LΛ is onto and therefore L is onto. Further, for each h, k H < Λ ω Lh, Λ ω Lk > dµ(ω) = < SLh, Lk >=< L SLh, k >. Thus the cg-frame operator of {Λ ω L} ω is L SL. If K is an invertible operator on H then the ranges of analysis operators for the given cg-frame {Λ ω } ω and the cg-frame {Λ ω K} ω coincide. Corollary 4.2. If {Λ ω } ω is a cg-frame for H and L : H H is a bounded operator, then {Λ ω + Λ ω L} ω is a cg-frame for H if and only if I + L is onto. In this case, the cg-frame operator for {Λ ω + Λ ω L} ω is (I + L) S(I + L) and the frame bounds are (I + L) 2 A, I + L 2 B.

11 g-frame sequence operators, cg-riesz bases 3367 In particular, if L is a positive operator (or just I + L > ɛ, for some ɛ > 0) then {Λ ω + Λ ω L} ω is a cg-frame with cg-frame operator S + L S + SL + L SL. Corollary 4.3. If {Λ ω } ω is a cg-frame for H and P is an orthogonal projection on H, then for all a 1, {Λ ω + aλ ω P } ω is a cg-frame for H. Proposition 4.4. Let (, µ) be a measure space where µ is σ-finite. Let {Λ ω } ω and {θ ω } ω be cg-bessel families for H with respect to {H ω } ω and with synthesis operators T 1, T 2 and cg-frame operators S 1, S 2, respectively. For the given operators L 1, L 2 : H H the following are equivalent: (i) {Λ ω L 1 + θ ω L 2 } ω is a cg-frame for H. (ii) T1 L 1 + T2 L 2 is a bounded operator on H, which is bounded below. (iii) S = L 1S 1 L 1 + L 2S 2 L 2 + L 1T 1 T2 L 2 + L 2T 2 T1 L 1 > ɛ, for some ɛ > 0. Moreover, in this case, S is the cg-frame operator of {Λ ω L 1 + θ ω L 2 } ω. Proof. (i) (ii) {Λ ω L 1 + θ ω L 2 } ω is a cg-frame if and only its analysis operator T is a bounded and bounded below operator, in which T h(ω) = {(Λ ω L 1 + θ ω L 2 )h} ω = {Λ ω L 1 h} ω + {Λ ω L 2 h} ω =T 1 L 1 h(ω) + T 2 L 2 h(ω), h H, ω. (ii) (iii) Let T be the synthesis operator of cg-bessel family {Λ ω L 1 + θ ω L 2 } ω. The cg-frame operator of {Λ ω L 1 + θ ω L 2 } ω is S = T T = (T 1 L 1 + T 2 L 2 ) (T 1 L 1 + T 2 L 2 ) = L 1S 1 L 1 + L 2S 2 L 2 + L 1T 1 T 2 L 2 + L 2T 2 T 1 L 1. So T is bounded below if and only if S > ɛ, for some ɛ > 0. Theorem 4.5. Let {Λ ω } ω be a cg-frame for H with synthesis operator T 1 and cg-frame operator S 1 and {θ ω } ω be a cg-bessel family in H with synthesis operator T 2 and cg-frame operator S 2. Suppose that ranget2 ranget1. If the operator R = T 1 T2 is a positive operator, then {Λ ω + θ ω } ω is a cg-frame for H with cg-frame operator S 1 + R + R + S 2. Proof. Let L 1 = I = L 2, by Proposition 4.4, the cg-frame operator of {Λ ω + θ ω } ω is S = S 1 + S 2 + T 1 T 2 + T 2 T 1 = S 1 + S 2 + R + R. Corollary 4.6. If {Λ ω } ω is a cg-frame for H with cg-frame operator S and {θ ω } ω is a cg-bessel family for H such that < h, k >= < Λ ω h, θ ω k > dµ(ω), h, k H, then for all real numbers a and b, {Λ ω S a + θ ω S b } ω is a cg-frame for H.

12 3368 M. Madadian and M. Rahmani Proof. For each h, k H, < S a+b h, k >=< S a h, S b k >= < Λ ω S a h, θ ω S b k > dµ(ω) = < {Λ ω (S a h)} ω, {θ ω (S b k)} ω >=< T 1 h, T2 k >, where T 1 and T 2 are the synthesis operators of cg-frames {Λ ω S a } ω and {θ ω S b } ω, respectively. So S a+b = T 1 T2 = R. Therefore {Λω S a + θ ω S b } ω is a cg-frame by Theorem 4.5. Corollary 4.7. If {Λ ω } ω is a cg-frame for H with cg-frame operator S and {θ ω } ω is a dual cg-frame of f then for all real numbers a and b, {Λ ω S a + θ ω S b } ω is a cg-frame for H. Proposition 4.8. Let {Λ ω } ω be a cg-frame for H with cg-frame operator S and frame bounds A and B. Let { 1, 2 } be a partition of such that 1 and 2 are measurable. Let S j be the cg-frame operator of cg-bessel family {Λ ω } ω j, j = 1, 2. Then for all real numbers a and b, the family is a cg-frame for H. {θ ω } ω = {Λ ω + Λ ω S a } ω 1 {Λ ω + Λ ω S b } ω 2 Proof. Let a, b R, then for each h H, ( (Λ ω + Λ ω S a )h 2 dµ(ω)) ( Λ ω h 2 dµ(ω)) ( Λ ω S a h 2 dµ(ω)) similarly, B h + B S a 1h B(1 + S a 1 ) h, ( (Λ ω + Λ ω S a )h 2 dµ(ω)) 1 2 B(1 + S b 2 ) h. 2 Thus {θ ω } ω is a cg-bessel family. The cg-frame operator of {Λ ω + Λ ω S a } ω 1 (I + S a 1)S 1 (I + S a 1) = S 1 + 2S 1+a 1 + S 1+2a 1 S 1, similarly for {Λ ω + Λ ω S b } ω 2. Hence, S, the cg-frame operator of {θω } ω satisfies S S 1 + S 2 = S > 0. Therefore, {θ ω } ω is a cg-frame for H. is

13 g-frame sequence operators, cg-riesz bases 3369 References [1] M.R. Abdollahpour and M.H. Faroughi, Continuous G-Frames in Hilbert spaces, Southeast Asian Bulletin Mathematics, 32 (2008), [2] P. Balazs and M.A. El-Gebeily, A Systematic Study of Frame Sequence Operators and their Pseudoinverses, Int. Math. Forum., 3(5) (2008), [3] O. Christensen, Frames and Bases: An Introductory Course. Birkhauser, Boston, [4] I. Daubechies and A. Grossmann and Y. Meyer, Painless nonorthogonal Expansions. J. Math. Phys., 27(5) (1986), [5] R.J. Duffin and A.C. Schaeffer, A class of nonharmonik Fourier series. Trans. Amer. Math. Soc., 72(1) (1952), [6] A. Khosravi and K. Musazadeh, Fusion frames and g-frames, J. Math. Anal. Appl., 342 (2008), [7] S. Obeidat, S. Samarah, P.G. Casazza and J.C. Tremain, Sums of Hilbert Space Frames, J. Math. Anal. Appl., 351 (2009), [8] W. Rudin, Functional Analysis. MacGraw-Hill, New York, [9] W. Sun, G-frames and G-Riesz bases, J. Math. Anal. Appl., 322 (2006), Received: April, 2010

Duals of g-frames and g-frame Sequences

International Mathematical Forum, Vol. 8, 2013, no. 7, 301-310 Duals of g-frames and g-frame Sequences Mostafa Madadian Department of Mathematics, Tabriz Branch Islamic Azad University, Tabriz, Iran madadian@iaut.ac.ir