Duals of g-frames and g-frame Sequences

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1 International Mathematical Forum, Vol. 8, 2013, no. 7, Duals of g-frames and g-frame Sequences Mostafa Madadian Department of Mathematics, Tabriz Branch Islamic Azad University, Tabriz, Iran Morteza Rahmani Department of Mathematics University of Tabriz, Tabriz, Iran m rahmani@tabrizu.ac.ir Abstract In this paper we study duals of g-frames and g-frame sequences. We define oblique dual g-frames and characterize these kind of duals. Also, we introduce Type I and Type II duals for g-frame sequences and verify some properties of them. Then, we state some conditions for existence of these duals. Mathematics Subject Classification: 42C15, 47A05 Keywords: g-frame, g-frame sequence, oblique projection, oblique dual 1 Introduction The concept of frames (discrete frames) in Hilbert spaces has been introduced by Duffin and Schaeffer ([6]) in 1952 to study some deep problems in nonharmonic Fourier series. After the fundamental paper [5] by Daubechies, Grossmann and Meyer, frame theory began to be 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 we generalize some results in [3] and [4] to g-frames and g-frame sequences. Throughout this paper, H is a Hilbert space and {H i } is a family of Hilbert spaces.

2 302 M. Madadian and M. Rahmani 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 g-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 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 Λ : H H

3 Duals of g-frames and g-frame sequences 303 S Λ = T Λ T Λ f = Λ i Λ if, (1.4) which is a bounded, self-adjoint, positive and invertible operator and where I is the identity operator on H. AI H S Λ BI H, Lemma 1.3. ([2]) Let H, K be Hilbert spaces, and suppose that U : K H is a bounded operator with closed range. Then there exists a bounded operator U : H K for which ker(u )=range(u), range(u )=ker(u), UU f = f, f range(u). We call the operator U the pseudo-inverse of U. This operator is uniquely determined by these properties. Lemma 1.4. ([2]) Let U : K H be a bounded operator with closed range. Then the following holds: (i) The orthogonal projection of H onto range(u) is given by UU. (ii) The orthogonal projection of K onto range(u ) is given by U U. (iii) U has closed range, and (U ) =(U ). (iv) On range(u), the operator U is given explicitly by U = U (UU ) 1. Given closed subspaces W and V such that H = W V (a direct sum, not necessarily orthogonal), the oblique projection of H on W along V is defined by E WV (ω + υ )=ω, ω W, υ V. The definition implies that range(e WV )=W and ker(e WV )=V. The orthogonal projection of H onto a subspace W will be denoted by P W. Note that E 2 WV = E WV. As a consequence of Lemma 2.1 in [3], the oblique projection E VW is also well defined. Straightforward calculation gives that the adjoint operator associated to the bounded operator E WV is 2 Oblique dual g-frames E WV = E VW. (1.5) We start this section by definition of dual g-frames and oblique dual g-frames. Definition 2.1. Let Λ={Λ i } be a g-frame sequence and Θ={Θ i } a g-bessel sequence in H and H Λ = span{λ i (H i)}. {Θ i } is an alternative dual of {Λ i }, or simply a dual, if T Λ T Θ HΛ = I HΛ.

4 304 M. Madadian and M. Rahmani Definition 2.2. Let {Λ i } be a g-frame for a closed subspace W of H and {Θ i } a g-frame for closed subspace V of H such that H = W V. {Θ i } is an oblique dual g-frame of {Λ i } on V when f = Λ i Θ i (f), f W. Lemma 2.3. Assume that {Λ i } and {Θ i } are g-bessel sequences in H and let W = span{λ i (H i )}, V = span{θ i (H i )}. Suppose that H = W V, then the following are equivalent: (i) f = Λ i Θ i(f), f W. (ii) E WV = Λ i Θ i. (iii) E VW = Θ i Λ i. (iv) E VW f,g = Λ i(f), Θ i (g), f,g H. (v) E WV f,g = Θ i(f), Λ i (g), f,g H. In case that equivalent conditions are satisfied, {Θ i } is an oblique dual g-frame of {Λ i } on V and {Λ i } is an oblique dual g-frame of {Θ i } on W. Furthermore, {Λ i } and {Θ i P W } are dual g-frames for W (in sense of classical definition) and {Θ i } and {Λ i P V } are dual g-frames for V. Proof. (i) (ii): For each f H we can write f = w + v such that w W and v V. We have E WV f = w = Λ i Θ i (w) = Λ i Θ i (w)+ Λ i Θ i (v )= Λ i Θ i (f). (ii) (i): It is obvious. (ii) (iii): Now, Let T and U denote the synthesis operators of {Λ i } and {Θ i }, respectively. So by (ii), TU = E WV. By (1.5), UT = E VW, so (iii) is shown. (iii) (ii): It is clear by (1.5). (iii) (iv): It is obvious. (ii) (v): It is obvious. (iv) (iii): Let f H. Since {Λ i } and {Θ i } are g-bessel sequences, Θ i Λ i is well defined. By (iv) we have E VW (f) Θ i Λ i(f),g =0,g H. Therefore, (iii) follows. Similarly, (v) implies (ii). If the equivalent conditions are satisfied, by (iv) and since E VW f = f, f V, we have f 2 = Λ i (f), Θ i (g) = Λ i P V (f), Θ i (g), f V.

5 Duals of g-frames and g-frame sequences 305 So, f 2 = Λ i P V (f), Θ i (g) Λ i P V (f) 2 Θ i (f) 2,f V. Since {Θ i } (respectively {Λ i P V } )isag-bessel sequence in H, {Λ i P V } (respectively {Θ i } ) satisfies the lower g-frame condition for all f V. Therefore both are g-frames for V. Also, by (iii) they are dual g-frames in the classical sense. Similarly, {Λ i } and {Θ i P W } are dual g-frames for W. By (ii) and (iii) and Definition 2.2, the statement about the relevant g-frames being oblique duals follows. Proposition 2.4. Let {Λ i } and {Θ i } be g-bessel sequences in H and f = Λ i Θ i(f), f W. Let U be any closed subspace of H for which H = W U. Then {Θ i E WU } is an oblique dual g-frame of {Λ i } on U. Proof. By assumption, we have f = Λ i Θ i E WU f, f W. Also, for each f U, f,e UW Θ i Λ i g =0,g H. Then Thus f,e UW Θ i Λ i g =0,f U,g H. E WU f = Λ i Θ i E WU f, f H. (2.1) Let T and V be the synthesis operators of {Λ i } and {Θ i }, respectively. By (2.1), E WU = TV E WU, (2.2) equivalently, E UW = E UW VT. For each f H, we have E UW f = E UW Θ i Λ i f,

6 306 M. Madadian and M. Rahmani in particular, f = E UW Θ i Λ if, f U. Then for each f U, f 2 = E UW Θ i Λ if,f = Λ i f,θ i E WU f. (2.3) By (2.3), {Θ i E WU } satisfies the lower g-frame condition on U. Hence it is a g-frame for U. Via (2.2), it is clear that {Θ i E WU } is an oblique dual g-frame of {Λ i } on U. 3 Types I and Type II duals of g-frame sequences In this section, we introduce the Type I and Type II duals of a g-frame sequence and then verify some properties of these types of duals. Definition 3.1. Let Λ={Λ i } be a g-frame sequence and Θ={Θ i } a g-bessel sequence in H. (i) {Θ i } is a Type I dual of {Λ i },if{θ i } is a dual and range(t Θ ) range(t Λ ). (ii) {Θ i } is a Type II dual of {Λ i },if{θ i } is a dual and range(t Θ ) range(t Λ ). Lemma 3.2. Let {Λ i } be a g-frame sequence with canonical dual g-frame sequence { Λ i } (for more details about the canonical dual g-frame sequence, see [8]). Then: (i) H Λ = span{ Λ i (H i)} = range(t Λ )=range(t Λ) =span{λ i (H i)} = H Λ. (ii) T Λ = T Λ S Λ = T Λ (T ΛT Λ ). (iii) range(t Λ )=range(t Λ) and this is a closed subspace of H. Proof. (i). We have { Λ i } = {Λ i S Λ }, where S Λ = T Λ TΛ. By Corollary 2.3 and Lemma 2.6 in [8], H Λ = span{ Λ i (H i)} = H Λ = span{(s Λ ) Λ i (H i)} =(S Λ ) span{λ i (H i )} =(S Λ ) (V )=V.

7 Duals of g-frames and g-frame sequences 307 By Proposition 2.1 and Corollary 2.5 in [8], range(t Λ )=H Λ. (ii). By Lemma 2.6 and Corollary 2.3 and Proposition 2.1 and Corollary 2.2 in [8], we have T Λ = U Λ = ŨΛ = US 1 Λ P = Uι V S 1 Λ P = UPι V S 1 P = TΛ S, where ι V : V H is the inclusion operator and U Λ, Ũ Λ and S Λ are defined in [8]. (iii). By Proposition 2.1(iv) and Corollary 2.2(iii) in[8], where U Λ and S Λ are defined in [8]. range(t Λ) =range(u Λ) =range(u Λ) = range(u Λ )S 1 Λ = range(u Λ) = range(u Λ )=range(t ), Theorem 3.3. Let Λ={Λ i } be a g-frame sequence and Θ={Θ i } a g-bessel sequence in H. (i) If Θ is a Type I dual of Λ then Θ is a g-frame sequence and Λ is a Type I dual of Θ. In this case, Λ and Θ are oblique duals, H Θ = range(t Θ )=range(t Λ )= H Λ and H = range(t Λ ) range(t Θ ). (ii) If Θ is a Type II dual of Λ then Θ is a g-frame sequence and Λ is a Type II dual of Θ. In this case, Λ and Θ are oblique duals, range(t Θ )=range(t Λ ) and Proof. (i). By assumption, we have H = range(t Λ ) range(t Θ ). f = T Λ T Θ(f) = Λ i Θ i (f), f H Λ. So for each f H Λ, f 2 = Θ i f,λ i f ( Θ i f 2 ) 1 2 ( Λ i f 2 ) 1 2. Since Λ is a g-frame sequence, so Θ is a g-frame sequence. Also, T Θ T Λ HΘ = I HΘ.

8 308 M. Madadian and M. Rahmani Therefore Λ is a Type I dual of Θ. The other parts are obvious. (ii). Let Λ be the canonical dual g-frame sequence of Λ. Then TΘ = P range(t Θ )TΘ = P range(t Λ )TΘ (since Θ is a Type II dual) = T Λ T ΛTΘ (Lemma 1.3 and Lemma 1.4) = T Λ (since Θ is a dual of Λ) = T Λ T ΛT Λ (since Λ is a dual of Λ) = P range(t Λ )T Λ (Lemma 1.3 and Lemma 1.4). Hence, TΘ H Λ = T Λ H Λ. Now, since TΛ has closed range (by Corollary 2.5 in [8]), TΛ : H Λ = range(t Λ ) range(tλ ) is a bijection. Likewise is a bijection. Therefore Hence T Λ : H Λ = range(t Λ) range(t Λ ) range(tλ)=range(t Λ) = range(t Λ H Λ ) = range(tθ HΛ ) range(tθ ) range(t Λ ) (since Θ is a Type II dual). range(t Θ )=range(t Λ ) is closed, so range(t Θ ) is closed. Consequently, by Lemma 1.2 in [7], Θ is a g-frame sequence in H and H Θ = span{θ i (H i)} = range(t Θ ). Now, we claim that T Λ TΘ = P H Λ,HΘ. Since T ΛTΘ = I H Λ and range(t Λ )=H Λ, we have both (T Λ TΘ )2 = T Λ TΘ and range(t ΛTΘ )=H Λ. If f ker(t Λ TΘ ) then T ΛTΘ (f) = 0, so T Θ f range(t Θ ) ker(t Λ)=range(T Λ ) ker(t Λ)=0. Thus ker(t Λ TΘ ) ker(t Θ ) and the opposite inclusion is trivial. Therefore ker(t Λ T Θ)=ker(T Θ)=range(T Θ) = H.

9 Duals of g-frames and g-frame sequences 309 So T Λ TΘ = P H Λ,HΘ, also the fact that this oblique projection exists implies that H = range(t Λ ) range(t Θ ). Theorem 3.4. Let U and V be closed subspaces of H and {Λ i } be a g-frame for U. Then the following statements are equivalent: (i) H = U V. (ii) There is a g-frame {Θ i } for V which is a Type II dual of {Λ i }. (iii) There is a g-frame {Θ i } for V which is an oblique dual of {Λ i }. In case these hold, {Λ i P } is a g-frame for V and we can take {Θ i } to be the canonical dual g-frame of {Λ i P } in V, where P = P V U. Proof. (i) (ii): Suppose that H = U V and {Λ i } be a g-frame for U. We will construct a g-frame for V that is a Type II dual of {Λ i }. Recall from Proposition 2.1 in [4] that P := P V U : U V is an isomorphism, so P 1 V U is also an isomorphism. Therefore, {Λ i P } is a g-frame for V. Let {Θ i } be the canonical dual g-frame of {Λ i P } in V, that is, {Θ i } = {Λ i S 1 }. Given any u U we have P V u = Pu. Therefore, Λ i Θ i(u) =P 1 ( P Λ i Θ i(p V u)) = P 1 ( (Λ i p) Θ i (Pu)) = P 1 P (u). Hence {Θ i } is a dual of {Λ i } and furthermore {Λ i } is a g-frame for V. It remains to show that range(t( Θ ) range(t Λ ). Since range(t Λ) U, ifwe consider T Λ to be a mapping of i) H into U then the synthesis operator of ΛP = {Λ i P } is T ΛP = P T Λ. The corresponding g-frame operator l 2 S ΛP : V V for ΛP is S ΛP = T ΛP (T ΛP ) = P T Λ T Λ P and this is an invertible mapping of V onto itself. Therefore the canonical dual g-frame of ΛP is given by {Θ i } = {Λ i PS 1 ΛP } = {Λ i S 1 Λ P 1 }. Further, by Lemma 3.2, the synthesis operator for {Θ i } is T Θ = S 1 ΛP T ΛP = P 1 S 1 Λ (P ) 1 P T Λ = P 1 S 1 Λ T Λ. Since S Λ and P are both invertible, we conclude that ker(t Θ )=ker(t Λ ) and consequently range(t Θ )=range(t Λ ). (ii) (iii): This is obvious via Theorem 3.3.

10 310 M. Madadian and M. Rahmani (iii) (i): Suppose {Θ i } is an oblique dual of {Λ i } such that {Θ i } is also a g-frame for V. Set T = T Λ TΘ. Since {Θ i} is dual to {Λ i } and U = H Λ = range(t Λ ), we have T U = I U. We show that T = P U,V. By definition of dual, we have T 2 = T and range(t )=U. Also, since {Θ i } is an oblique dual, {Λ i } is dual to {Θ i } and therefore T V = T Λ TΘ V = I V. Hence range(t )=V. So ker(t )=V. This shows that T = P U,V and therefore, H = U V. Acknowledgements The authors would like to thank Tabriz Branch, Islamic Azad University for the financial support of this research, which is based on a research project contract. References [1] M.R. Abdollahpour and M.H. Faroughi, Continuous G-Frames in Hilbert spaces, Southeast Asian Bull. Math., 32 (2008), [2] O. Christensen, Frames and Bases: An Introductory Course. Birkhauser, Boston, [3] O. Christensen and Y.C. Eldar, Oblique dual frames and shift-invariant spaces. Appl. Comput. Harmon. Anal., 17 (2004), [4] C. Heil, Y.Y. Koo and J.k. Lim, Duals of Frame Sequences. Acta. Appl. Math., 107 (2009), [5] I. Daubechies and A. Grossmann and Y. Meyer, Painless nonorthogonal Expansions. J. Math. Phys., 27(5) (1986), [6] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72 (1952), [7] A. Khosravi and K. Musazadeh, Fusion frames and g-frames, J. Math. Anal. Appl., 342 (2008), [8] M. Madadian and M. Rahmani, g-frame sequence operators, cg-riesz bases and sum of cg-frames, International Mathematical Forum, 68(6) (2011), [9] W. Sun, G-frames and G-Riesz bases, J. Math. Anal. Appl., 322 (2006), Received: October, 2012

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

International Mathematical Forum, Vol. 6, 2011, no. 68, 3357-3369 g-frame Sequence Operators, cg-riesz Bases and Sum of cg-frames M. Madadian Department of Mathematics, Tabriz Branch, Islamic Azad University,