DUALITY PRINCIPLE IN g-frames

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1 Palestine Journal of Mathematics Vol. 6(2)(2017), Palestine Polytechnic University-PPU 2017 DUAITY PRINCIPE IN g-frames Amir Khosravi and Farkhondeh Takhteh Communicated by Akram Aldroubi MSC 2010 Classications: Primary 42C15; Secondary 42A38, 41A58. Keywords and phrases: g-orthonormal basis, g-riesz basis, g-riesz dual sequence. The authors express their gratitude to the referee for very valuable suggestions. Abstract. The concept of Riesz dual sequences (R-dual sequences) was introduced by Casazza et al. in Recently, for generalizing this concept to g-frames the concept of g- Riesz dual sequences has been introduced and various denitions of R-duals for frames are in the literature. In this paper, we generalize these concepts for g-frame and introduce g-riesz duals (g-r-duals) of type II, III and IV. Since the g-r-dual of type IV is the most general g-r-dual, we focus on the g-r-dual of type IV. We give characterizations of g-frames and g-riesz bases in terms of their g-r-dual of type IV. We characterize all dual g-frames of a g-frame in terms of its g-r-dual of type IV which can be considered as Wexler-Raz biorthogonality relations for g-frames. Also, we present a generalization of Ron-Shen duality principle to g-frames. In addition, we investigate the construction of dual g-frames in more details and we give another characterization of dual g-frames with respect to its g-r-dual sequence. 1 Introduction and preliminaries The concept of R-duality of a Bessel sequence in a separable Hilbert space was introduced by Casazza, Kutyniok and ammers in [1], in order to obtain a generalization of duality principles in Gabor frames to abstract frame theory. et (e i ), (h i ) be orthonormal bases and (f i ) be a Bessel sequence. The R-dual sequence of (f i ) with respect to the orthonormal bases (e i ) and (h i ) is the sequence (w f j ), such that for every j I w f j = f i, e j h i. The R-duality has been favored by many authors. The R-duality with respect to orthonormal bases has been discussed in [2] and [3]. In [8], the authors introduced various alternative R-duals and showed their relation with Gabor frames. In [11], the authors generalized the R-duality in Banach spaces. In [4] the authors, proved that the duality principle extends to any pair of projective unitary representation of countable groups. Recently, for generalizing this concept to g-frames the concept of g-riesz dual sequences has been introduced [7]. Various denitions of R-duals for frames are in the literature. In this paper, we generalize these concepts to g-frame and introduce g-riesz duals (g-rduals) of type II, III and IV. Since the g-r-dual of type IV is the most general g-r-dual, we focus on the g-r-dual of type IV. We give characterizations of g-frames and g-riesz bases in terms of their g-r-dual of type IV. We characterize all dual g-frames of a g-frame in terms of its g-r-dual of type IV, which can be considered as Wexler-Raz biorthogonality relations for g-frames. Also, we present a generalization of Ron-Shen duality principle to g-frames. In addition, we investigate the construction of dual g-frames in more details and we give another characterization of dual g-frames with respect to its g-r-dual sequence. Throughout this paper H denotes a separable Hilbert space with inner product.,. and I is a subset of Z, and {H i : i I} is a sequence of separable Hilbert spaces. Also, for every i I, (H, H i ) is the set of all bounded, linear operators from H to H i.

2 404 Amir Khosravi and Farkhondeh Takhteh In the rest of this section we review several well-known denitions and results. The new results are stated in Section 2. For every sequence {H i }, the space ( Hi ) l 2 = {(f i ) : f i H i, i I, f i 2 < } with pointwise operations and the following inner product is a Hilbert space (f i ), (g i ) = f i, g i. A sequence = { i (H, H i ) : i I} is called a g-frame for H with respect to {H i : i I} if there exist A, B > 0 such that for every f H A f 2 i f 2 B f 2, A, B are called g-frame bounds. We call a tight g-frame if A = B and a Parseval g-frame if A = B = 1. If only the right hand side inequality is required, is called a g-bessel sequence. If is a g-bessel sequence, then the synthesis operator for is the linear operator, T : ( Hi ) l 2 H T (f i ) = i f i. We call the adjoint of the synthesis operator, the analysis operator. The analysis operator is the linear operator, T : H ( Hi ) l 2 T f = ( i f). We call S = T T the g-frame operator of and S f = i if, If = ( i ) is a g-frame with lower and upper g-frame bounds A, B, respectively, then the g-frame operator of is a bounded, positive and invertible operator on H and so A f, f S f, f B f, f (f H) A.I S B.I. The canonical dual g-frame for = ( i ) is dened by = ( i ), where i = i S 1 which is also a g-frame for H with lower and upper g-frame bounds 1 B and 1 A, respectively. Also for every f H, we have f = i i f = i i f. We say = { i (H, H i ) : i I} is a g-frame sequence if it is a g-frame for span{ i (H i)}. A sequence = { i (H, H i ) : i I} is g-complete if {f : i f = 0, i I} = {0}. We note that the g-bessel sequence is g-complete if and only if T is injective. We say that is a g-orthonormal basis for H, if and i f i, j f j = δ i,j f i, f j, f i H i, f j H j, i, j I i f 2 = f 2 A sequence = { i (H, H i ) : i I} is a g-riesz sequence if there exist A, B > 0 such that for every nite subset F I and g i H i, i F A i F g i 2 i F i g i 2 B i F g i 2. (1.1)

3 DUAITY PRINCIPE IN g-frames 405 G-Riesz sequence = { i (H, H i ) : i I} is called a g-riesz basis if it is g-complete, too. So is a g-riesz basis if and only if T is a bounded invertible operator. Clearly, every g-orthonormal basis is a g-riesz basis. et = { i (H, H i ) : i I} and Q = {Q i (H, H i ) : i I} be g-bessel sequences with g-bessel bounds B and C, respectively. The operator S Q : H H dened by S Q f = i Q if, (f H) is a bounded operator, S Q BC, S Q = S Q and S = S. Two g-bessel sequences = { i (H, H i ) : i I} and Q = {Q i (H, H i ) : i I} are called dual g-frames if f = i Q if = Q i if, For more details about g-frames, see [6, 9]. 2 Main results In this section, rst we consider the g-riesz dual(g-r-dual) with respect to g-orthonormal bases as the g-r dual of type I in [7] and we introduce alternative denitions of g-r-duals. Denition 2.1. et = { i (H, H i ) : i I} be a g-frame for H with g-frame operator S. (i) et G = {G i (H, H i ) : i I} and = { i (H, H i ) : i I} be g-orthonormal bases. The g-r-dual of type I of with respect to G and is F = (F j ) dened by F j f = G j S f (ii) et G = {G i (H, H i ) : i I} and = { i (H, H i ) : i I} be g-orthonormal bases. The g-r-dual of type II of with respect to G and is F = (F j ) dened by F j f = G j S 1 2 S( S 1 f 2 ) (iii) et G = {G i (H, H i ) : i I} and = { i (H, H i ) : i I} be g-orthonormal bases and M : H H be a bounded invertible operator with M S and M 1 S 1. The g-r-dual of type III of with respect to triplet (G,, M) is F = (F j ) dened by F j f = G j S (S 1 2 )( M) f (iv) et G = {G i (H, H i ) : i I} and = { i (H, H i ) : i I} be g-riesz bases. The g-r-dual of type IV of with respect to G and is F = (F j ) dened by F j f = G j S f In all of the above cases, it is obvious that F j is well-dened and F j (H, H j), for every j I. Clearly, the g-r-duals of type II are contained in the class of g-r-duals of type III and the g-r-duals of type III are contained in the class of g-r-duals of type IV. Moreover, the g-rduals of type I, II, and III are contained in the class of g-r-duals of type IV. In the following proposition, we show that for tight g-frames the g-r-duals of type I, II and III coincide.

4 406 Amir Khosravi and Farkhondeh Takhteh Proposition 2.2. et = { i (H, H i ) : i I} be a tight g-frame. Then the g-r-duals of type I, II and III coincide. Proof. Denote the g-frame operator for by S. Since is a tight g-frame, then S = AI for some A > 0. For every j I, G j S = G j A 1 2 S = 1 ( A 2 ) G js 1 2 S 1. Therefore the g-r-duals of type I ( S 2 ) and II coincide. et G = {G i (H, H i ) : i I} and = { i (H, H i ) : i I} be g-orthonormal bases. Take the bounded invertible operator M : H H such that M S = A and M 1 S 1 = 1 A. Suppose that (g i ) ( H i) l 2, then we have ( i M) g i 2 = M i g i 2 M 2 i g i 2 A g i 2. and ( i M) g i 2 = M 1 i g i 2 (M ) 1 2 i g i 2 A g i 2. Therefore ( i M) is a tight g-riesz basis with bound A. We can see that M A is a unitary operator. Since is a g-orthonormal basis, then ( 1 A i M) is a g-orthonormal basis, denote it by (Y i ). Hence ( AY i ) = ( i M). Now, the g-r-dual of of type III with respect to (G,, M) is F j = G j S (S 1 2 )( M) = G js ( 1 A )( AY) = G js ()(Y) which is a g-r-dual of type I of. Since the g-r-duals of type II are contained in the class of g-r-duals of type III and for tight g-frames the g-r-duals of type I and II coincide, then for tight g-frames, g-r-duals of type I, II and III coincide. Since the g-r-dual of type IV is the most general g-r-dual, we focus on the g-r-dual of type IV and we give some characterizations of it. Note that all results about the g-r-dual of type IV hold for the g-r-duals of type I, II and III. In the following proposition, we present an algorithm which invert the process of mapping to its g-r dual of type IV (F ). Proposition 2.3. et = { i (H, H i ) : i I} be a g-bessel sequence with g-bessel bound A and G = {G i (H, H i ) : i I} and = { i (H, H i ) : i I} be g-riesz bases. et F = (F j ) be the g-r dual sequence of type IV of with respect to G and. Then F is a g-bessel sequence and is the g-r-dual sequence of type IV of F with respect to g-riesz bases ( i ) and ( G i ), where ( i ) and ( G i ) are dual g-riesz bases of and G, respectively. In the sense that for every i I we have i f = i F Gj j f = i S F Gf Proof. et B and C be upper g-riesz bounds for G and, respectively. Since is a g-riesz basis with upper g-riesz bound C, then it is a g-frame with upper g-frame bound C, too. On the other hand, is a g-bessel sequence with g-bessel bound A. Therefore S AC, see [6]. Hence for every f H we have F j f 2 = Therefore F is a g-bessel sequence in H. G j S f 2 B S f 2 ABC f 2.

5 DUAITY PRINCIPE IN g-frames 407 For every f H and g i H i we have i S Gf, F g i = F Gj j f, i g i = S G j G j f, i g i = S G j G j f, i g i = f, S i g i = f, k I k k i g i = f, i g i = i f, g i. Thus for every i I i f = i F Gj j = i S F Gf Corollary 2.4. et = { i (H, H i ) : i I} be a g-frame with g-frame operator S and (F j ) be the g-r-dual of type III of with respect to g-orthonormal bases G = {G i (H, H i ) : i I}, = { i (H, H i ) : i I} and invertible operator M. Then for every i I and f H, i f = i (M ) 1 S (F)(GS 1 2 ) f. Also, if (F j ) is the g-r-dual of type II of with respect to g-orthonormal bases G and, then for every i I and f H, i f = i S 1 2 SF(GS 1 2 ) f. Proof. Since (F j ) is the g-r-dual of type III of with respect to g-orthonormal bases G and and the bounded invertible operator M, then (F j ) is the g-r-dual of type IV of with respect to g-riesz bases (G j S 1 2 ) and ( i M). By Proposition 2.3, we have i f = ( i M) S f. (G js 1 2 ) It is easy to check that that ( i M) = ( i (M ) 1 ) and (G j S 1 2 ) = (G j S 1 2 ). Therefore for every i I and f H, i f = i (M ) 1 S f. 1 (F)(GS 2 ) Since the class of g-r-duals of type II is contained in the class of g-r-dual of type III, by substituting M = S 1 2 in the above equation, we have i f = i S 1 2 S f. 1 F(GS 2 ) In the following theorem, we present an equivalent condition for the sequence = { i (H, H i ) : i I} to be a g-frame, which can be regarded as a generalization of Ron-Shen duality principle to g-frames. Theorem 2.5. et = { i (H, H i ) : i I} be a g-bessel sequence in H and F = {F j (H, H j ) : j I} be the g-r-dual sequence of type IV of with respect to g-riesz bases G = {G i (H, H i ) : i I} and = { i (H, H i ) : i I}. Then is a g-frame if and only if F is a g-riesz sequence. Proof. et 0 < B 1 B 2 and 0 < C 1 C 2 be g-riesz bounds for G and, respectively. Suppose that is a g-frame with bounds 0 < A 1 A 2. For every nite subset F I we have F j gj 2 = S G j g j 2 = S ( G j g j ) 2 A 2 C 2 G j g j 2 A 2 B 2 C 2 g j 2. Similarly, we can get the following result F j gj 2 A 1 B 1 C 1 g j 2.

6 408 Amir Khosravi and Farkhondeh Takhteh Therefore (F j ) is a g-riesz sequence in H. Conversely, let (F j ) be a g-riesz sequence with g-riesz bounds 0 < D 1 D 2 in H. Suppose that f span (G j H j), then there is a nite set F I and {g j H j : j F } such that f = G j g j. We have i f 2 = i ( G j g j ) 2 = i (G j g j ) 2 1 i C ig j g j 2 = 1 i 1 C ig j g j 2 1 = 1 F j gj 2 D 2 g j 2 C 1 C 1 D 2 B 1 C 1 G j g j 2 = D 2 B 1 C 1 f 2. Similarly, we can get the following result i f 2 D 1 B 2 C 2 f 2. Since span (G j H j) = H, then is a g-frame in H. In the following theorem, we give a characterization of g-riesz bases in terms of their g-rdual of type IV. Theorem 2.6. et = { i (H, H i ) : i I} be a g-bessel sequence for H and F = {F j (H, H j ) : j I} be the g-r-dual sequence of type IV of with respect to g-riesz bases G = {G i (H, H i ) : i I} and = { i (H, H i ) : i I}. Then = { i (H, H i ) : i I} is a g-riesz basis if and only if F is a g-riesz basis. Proof. We know that is a g-bessel sequence if and only if F is a g-bessel sequence. For every f H, we have S f = G j G j(s f) = G j F j f = S GF f. Therefore S = S GF. Since S = T T and is a g-riesz basis, then S is invertible if and only if T is invertible which is equivalent to is a g-riesz basis. Therefore is a g-riesz basis if and only if S is invertible. Similarly F is a g-riesz basis if and only if S F is invertible. G Since S GF = S F G by the above relation, is a g-riesz basis if and only if F is a g-riesz basis. We note that, since every g-orthonormal basis is a g-riesz basis, the above theorem is a generalization of Proposition 3.10 in [7]. In the following theorem, we characterize all dual g-frames of a g-frame in terms of its g-rdual of type IV which can be considered as a generalization of Wexler-Raz biorthogonality relations to g-frames. Theorem 2.7. et = { i (H, H i ) : i I}, Y = {Y i (H, H i ) : i I} be g-frames and G = {G i (H, H i ) : i I}, = { i (H, H i ) : i I} be g-riesz bases in H. et F Y be the g-r-dual of type IV of Y with respect to g-riesz bases G, and F be the g-r-dual of type IV of with respect to g-riesz bases G and. Then the following statements are equivalent: (i) Y and are dual g-frames. (ii) S Y = S Y = I. (iii) F Y j gj, F k g k = δ jk g j, g k g j H j, g k H k (j, k I).

7 DUAITY PRINCIPE IN g-frames 409 Proof. The equivalence of (1) and (2) is obvious. Since is a g-riesz basis, Corollary 3.3 in [9], easily implies that S Y S g j H j, g k H k, j, k I we have = S Y. For every Therefore F Y j F Y j gj, F k gk = S Y G j g j, S G k g k = G j g j, S Y G k g k. gj, F k g k = δ jk g j, g k if and only if G j g j, S Y ( G k g k) = G j g j, G k g k which is equivalent to S Y ( G k g k) = G k g k, for every k I. Since span G i (H i ) = H and S Y is continuous, this is equivalent to S Y = I. Therefore (2) is equivalent to (3). In the following lemma we prove that if is a g-bessel sequence, then there exists a basic relation between its synthesis operator and span F j (Hj ), see [7, emma 3.6]. emma 2.8. et = { i (H, H i ) : i I} be a g-bessel sequence with synthesis operator T and F = {F i (H, H i ) : i I} be the g-r-dual sequence of type IV of with respect to g-riesz bases G = {G i (H, H i ) : i I}, = { i (H, H i ) : i I}. et (h i ) ( H i) l 2 and h H. Then (i) h ker(t ) = F span F j (Hj ) if and only if ( i h) kert (equivalently S h = 0). (ii) (h i ) kert if and only if i h i ker(t ) = F span F j (Hj ). (iii) F is g-complete if and only if T is injective. Proof. (1) h ker(t ) = F span F j (Hj ) if and only if for every j I, g j H j, h, F j gj = 0. For every j I we have h, F j gj = h, S G j g j = S h, G j g j = i ih, G j g j. Since spang j (H j) = H, then for every j I, i ih, G j g j = 0 if and only if i ih = 0. Therefore, h span F j (Hj ) if and only if ( i h) kert. (2) et h = i h i. Since ( i ) is a g-riesz basis, then (h i ) = ( i h). Thus (h i ) kert if and only if i i h = 0, now by using (1) (h i ) kert if and only if h = i h i span F j (Hj ) = ker(t F ). (3) By (2), T is injective if and only if T F is injective and we know that F is g-complete if and only if T F is injective. Therefore, F is g-complete if and only if T is injective. In the following theorem, we give another characterization of dual g-frames. Theorem 2.9. et = { i (H, H i ) : i I} be a g-frame with g-frame operator S and F = {F i (H, H i ) : i I} be the g-r-dual sequence of type I of with respect to g- orthonormal bases G = {G i (H, H i ) : i I}, = { i (H, H i ) : i I}. Then the following statements are equivalent: (i) Q is a dual g-frame of. (ii) There exists a g-bessel sequence {Mj for every g j H j, j I F Q j ({span F j (Hj )}, H j ) : j I} such that gj F S 1 j g j = M j g j. Proof. et Q = (Q i ) be a dual g-frame of = ( i ). Then for every g j H j, j I, we have G j g j = S Q (G j g j ) = = = i Q ig j g j = i (Q i i S 1 )G j g j + i (Q i i S 1 i is 1 G j g j i (Q i i S 1 )G j g j + S S 1 G j g j. + is 1 )G j g j

8 410 Amir Khosravi and Farkhondeh Takhteh Since S S 1 = I, then i (Q i i S 1 )G j g j = 0 and by emma 2.8, we have i (Q i i S 1 )G j g j {span F j (Hj )}. Now, dene M j : H j {span F j (Hj )} H by Then M j g j = F Q j M j g j = gj F S 1 j g j i Q ig j g j (M j ) is a g-bessel sequence for {span F j i is 1 G j g j (g j H j, j I). (g j H j, j I). So Mj : {span F j (Hj )} H j and (Hj )} with respect to {H i ; i I}. Because, (Hj )}, we have let A be an upper g-frame bound for Q. Then for every f {span F j Mj f 2 = F Q j f F S 1 j f 2 = G j S Q f G j S S 1 f 2 = G j S Q f G j S 1 i if 2, since f {span F j (Hj )} by emma 2.8, i if = 0. Therefore G j S Q f 2 = S Q f 2 A f 2. M j f 2 = Conversely, suppose that (2) holds. Since for every g H, j I, G j g H j, then we have M j G j g = F Q j Therefore by [7, emma 3.3], for every i I So for every g l H l, l I we have Gj g F S 1 j G j g (Q i i S 1 )g = i M j G j g. i Q ig l g l = i ( i S 1 = G l g l + + Q i i S 1 )G l g l i (Q i i S 1 )G l g l = G l g l + i ( i M j (G j G l g l )) = G l g l + i i M j (G j G l g l ) = G l g l + i im l g l, since M l g l {span F j (Hj )}, then by emma 2.8, i im l g l = 0. Therefore Q is a dual g-frame of and this implies (1). Now, we present a characterization of the canonical dual g-frames. Corollary et = { i (H, H i ) : i I} be a g-frame with the canonical dual = { i (H, H i ) : i I} and F = {F j (H, H j) : j I} be the g-r-dual sequence of type I of with respect to g-orthonormal bases G = {G i (H, H i ) : i I} and = { i (H, H i ) : i I}. et Q = {Q i (H, H i ) : i I} be a dual g-frame of. Then for every g j H j, j I F Q j g j F j g j, with equality if and only if Q =.

9 DUAITY PRINCIPE IN g-frames 411 Proof. et T and T be the synthesis operators of and, respectively. Easily we can see that kert = kert, then by emma 2.8, span F j (H j ) = span F j (Hj ), so RanF j span F j (H j). On the other hand by the above theorem, for every g j H j, j I we have F Q j g j = F j g j + M j g j, where RanM j span F j (H j). But span F j (H j ) = span F j (Hj ), so RanM j span F j (H j). Then for every g j H j, j I we have F Q j g j 2 = F j g j 2 + M j g j 2 F j g j 2. By the above theorem, the equality holds if and only if Q =. References [1] P. Casazza, G. Kutyniok, and M.C. ammers, Duality principles in frame theory, J. Fourier Anal. Appl., 10, no. 4, (2004). [2] P. Casazza, G. Kutyniok, and M.C. ammers, Duality principle, localization of frames, and Gabor theory, Wavelets XI. Proceeding of the SPIE., 5914, (2005). [3] O. Christensen, H. O. Kim, and R. Y. Kim, On the duality principle by Casazza, Kutyniok, and ammers, J. Fourier Anal. Appl., 17, no. 4, (2011). [4] D. Dutkay, D. Han, D. arson, A duality principle for groups, J. Funct. Anal., 257, no. 4, (2009). [5] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, [6] A. Khosravi, K. Musazadeh, Fusion frames and g-frames, J. Math. Anal. Appl., 342, no. 2, (2008). [7] E. Osgooei, A. Najati, M. H. Faroughi, G-Riesz dual sequences for g-bessel sequences, Asian-Eur. J. Math., 7 no. 3, (15 pages) (2014). [8] D. T. Stoeva, O. Christensen., On R-duals and the duality principle in Gabor analysis, J. Fourier Anal. Appl., 21, no. 2, (2014). [9] W. Sun, G-frames and g-riesz bases, J. Math. Anal. Appl., 322, no. 1, (2006). [10] Y. C. Zhu, Characterization of g-frames and g-bases in Hilbert spaces, Acta Math. Sin., 24, no. 10, (2008). [11] X. M. Xiao, Y. C. Zhu, Duality principles of frames in Banach spaces, Acta. Math. Sci. Ser. A. Chin., 29, (2009). Author information Amir Khosravi and Farkhondeh Takhteh, Faculty of Mathematical Sciences and Computer, Kharazmi University, 599 Taleghani Ave., Tehran 15618, IRAN. khosravi amir@yahoo.com, khosravi@khu.ac.ir, ftakhteh@yahoo.com Received: July 7, Accepted: October 5, 2016.

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