On the Equality of Fusion Frames 1

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1 International Mathematical Forum, 4, 2009, no. 22, On the Equality of Fusion Frames 1 Yao Xiyan 2, Gao Guibao and Mai Ali Dept. of Appl. Math., Yuncheng University Shanxi , P. R. China Abstract In this paper, we discuss the property of fusion frames and fusion frames operators for Hilbert Spaces by utilizing the method of operator theory. Besides, we study the equality of fusion frames and obtain some meaningful results. Mathematics Subject Classification: 42C15, 47A55 Keywords: Hilbert Spaces, Frames, Fusion frames 1. Introduction Frames for Hilbert spaces were formally defined by Duffin and Schaeffer[1]in 1952 to study some deep problems in nonharmonic Fourier series. Basically, Duffin and Schaeffer abstracted the fundamental notion of Gabor frames for studying signal processing. Later general frame theory of subspaces introduces by P.Casazza and G.Kutyniok[2] as a natural generalization of the frame theory in Hilbert spaces. Since frame, in particular frame of subspaces(fusion frames), are applied to signal processing,image processing and data compression. In the paper, we consider fusion frames in a Hilbert space H, and give the equality for fusion frames. Let H be a Hilbert space. I be a set which is finite or countable, for every J I,J c = I \ J. We denote by B(H) the algebra of all bounded linear operators on H, and I H is the identity operator on H. 1 The Scientific Research Foundation for univerity of Shanxi (No ), Supported by the Emphasis Subject Foundation of Yuncheng University (No ). 2 yaoxiyan63@163.com

2 1060 Yao Xiyan, Gao Guibao and Mai Ali A sequence {f i } in a Hilbert space H is called a frame for H, if there exist 0 <A B< such that for all f H, A f 2 f,f i 2 B f 2. (1.1) The numbers A and B are called a lower and upper frame bound for the frame. A frame {f i } is called a tight frame if we chose A = B and a Parseval frame if A = B = 1. Let {f i } be a frame. Then the frame operator S(f) = f,f i f i associated with {f i } is a bounded, invertible and positive operator mapping H onto itself. This provides the reconstrction formula f = S 1 S(f) = f, f i f i = i f,f f i, (1.2) where f i = S 1 (f i ). The family { f i } ia also a frame for H, called the canonical dual frame of {f i }. Let {W i } be a family of closed subspaces of Hilbert space H and {v i } be a family of weights, i.e. v i > 0,i I. The family W = {(W i,v i )} is a fusion frame, if there exist constants 0 <C D<, such tnat C f 2 (f) 2 D f 2, f H, (1.3) where for the closed subspace W H, π W denote the orthogonal projection of H on W. The constants C and D are called the fusion frame bounds. The family {(W i,v i )} is called a C-tight fusion frame, if in (1.3) the constants C and D can be chosen so that C = D, a Parseval fusion frame provided that C = D =1. If we only have the upper bound, we call {(W i,v i )} a Bessel fusion sequence with Bessel fusion bound D. Let {(W i,v i )} be a fusion frame. Then the frame operaror S W for {(W i,v i )} is defined by S W f = (f), f H. The frame operator S W is a positive, self-adjoint, invertible operator on H with CI H S W DI H. Further we have the reconstruction formula [2]: f = vi 2 S 1 W π W i (f) = SW 1 (f), f H. (1.4) The family {(SW 1 W i,v i )} is called the dual fusion frame. If {(SW 1 W i,v i )} is a Bessel fusion sequence in H, for every J I we define the operator SW J by SW J f = (f), f H. (1.5) The main result of this paper deals with the equality of fusion frames in a Hilbert space H. We refer to [2,3,4] for an excellent introduction to fusion frames. Our references for frames are [1,5,6].

3 On the equality of fusion frames Main results In order to prove our assertions, we first give the following Lemma. Lemma 2.1 Let U, V be operators on H, and U + V = I H. Then U V = U 2 V 2. Proof. U V = (I V ) V =(I 2V + V 2 ) V 2 = (I V ) 2 V 2 = U 2 V 2. Now We discuss the equality of general fusion frames in H. Theorem 2.1 Let {(W i,v i )} be a fusion frame for H, and S W is the frame operaror for {(W i,v i )}. Then for all J I and all f H we have (f) 2 v 2 i π W i S 1 W (SJ W f) 2 = c v 2 i π W i (f) 2 v 2 i π W i S 1 W (SJc W f) 2. Proof. SinceS W = SW J + S J C W, it follows that SW 1 SW J + SW 1 S J C W = I H. Applying Lemma 2.1 to the two operators SW 1 SW J and SW 1 S J C W yields SW 1 SJ W S 1 W SJ W S 1 W SJ W = S 1 W SJC W S 1 W SJC W S 1 W SJC W. Further, for every f,g Hwe get SW 1 SW J f,g SW 1 SW J SW 1 SW J f,g = SW J f,sw 1 g SW 1 SW J f,sw J SW 1 g. (2.1) putting g = S W f and Applying (1.4),(1.5) obtains SW J f,sw 1 g SW 1 SW J f,sw J SW 1 g = SW J f,f SW 1 SW J f,sw J f = vi 2 π Wi (f) 2 v 2 i π Wi S 1 W (S J W f) 2. (2.2) Setting equality (2.2)equal to the corresponding equality for J c and using (2.1), we have SW J f,f S 1 W SJ W f,sj W f = (f) 2 SW 1 (SJc W f) 2. (2.3) c

4 1062 Yao Xiyan, Gao Guibao and Mai Ali Composing (2.2) and (2.3), we finally get vi 2 π Wi (f) 2 v 2 i π Wi S 1 W (S J W f) 2 = c v 2 i π Wi (f) 2 v 2 i π Wi S 1 W (S J c W f) 2. In the situation of Parseval fusion frames the equality is of special form. Theorem 2.2 Let {(W i,v i )} be a Parseval fusion frame for H. Then for all J I and all f H, we have (f) 2 (f) 2 = (f) 2 c c Proof. Let S W denote the frame operaror for {(W i,v i )}. Since {(W i,v i )} is a Parseval fusion frame, its frame operaror equals the identity operator, i.e.s W = I H. Employing Theorem 2.1 and the fact that {(W i,v i )} is a Parseval fusion frame obtains vi 2 π Wi (f) 2 vi 2 π Wi (f) 2 = vi 2 π Wi (f) 2 SW J f 2 = (f) 2 SW 1 S W f,sw J f = (f) 2 SW 1 (SJ W f) 2 = vi 2 π Wi (f) 2 vi 2 π Wi SW 1 (S J c W f) 2 c = vi 2 π Wi (f) 2 S J c W f 2 c = (f) 2 c c Remark 2.1 Each side of the Parseval fusion frame equality is nonnegative, bacause SW J has operator norm at most 1. Theorem 2.3 Let {(W i,v i )} be a Parseval fusion frame for H. For every J I, every E J c, and every f H, we have vi 2 π Wi (f) 2 vi 2 π Wi (f) 2 E c \E = (f) 2 (f) 2 +2 c i E

5 On the equality of fusion frames 1063 Proof. Appling Theorem 2.2 twice yields (f) 2 (f) 2 E c \E = vi 2 π Wi (f) 2 vi 2 π Wi (f) 2 E c \E = (f) 2 (f) 2 +2 (f) 2 c i E = (f) 2 (f) 2 +2 c i E Since each C-tight fusion frame can be turned into a Parseval fusion frame by a change of siae, we have the following corollary. Corollary 2.1 Let {(W i,v i )} be a C-tight fusion frame for H. Then for every J I, and every f H we have C (f) 2 (f) 2 = C (f) 2 c c Proof. Note that { 1 C W i,v i } is a Parseval fusion frame for H, because {W i,v i } is a C-tight fusion frame for H. Employing Theorem 2.2 prove the result. If {W i,v i } is a Parseval fusion frame for H, then for every J I, and every f H we have f 2 = (f) 2 + c Hence, one of two terms on the right-hand-side of the above equality is greater than or equal to 1 2 f 2. It follows from Theorem 2.2 that for every J I, and every f H, vi 2 π Wi (f) 2 + vi 2 π Wi (f) 2 = vi 2 π Wi (f) 2 + vi 2 π Wi (f) 2 1 c c 2 f 2. We will now see that actually the right-hand-side of this inequlity is in fact much larger. Proposition 2.1 Let {(W i,v i )} be a parseval fusion frame for H. Then for every J I, and every f H we have (f) 2 + (f) f 2. c

6 1064 Yao Xiyan, Gao Guibao and Mai Ali Futher, we have equlity for all f H if and only if SW J = 1I 2 H. Proof. By computing, we see that (S J W )2 +(S J c W )2 =2(S J W 1 2 I H) I H; and so (S J W ) 2 +(S J c W ) I H, with equality if and only if SW J = 1I 2 H. Since SW J + S J c W = I H, it follow that SW J +(S J c W ) 2 + S J c W +(SW J ) 2 3I 2 H. Applying Lemma 2.1 to U = SW J ang V = S J c W yields SW J +(S J c W ) 2 = S J c W +(SW J ) 2. Thus 2(SW J +(S J c W ) 2 )=SW J +(S J c W ) 2 + S J c W +(SW J ) 2 3I 2 H. Finally, for every f H we have (f) 2 + (f) 2 = SW J f,f + SJc W f,sj c W f = (SJ W +(SJc W )2 )f,f 3 4 I H. c From above, we clearly have equlity for all f H if and only if SW J = 1I 2 H. For proving the next Theorem 2.4, we need a result corcerning the operators SW J,S J c W. We first give the following Lemma 2.2. lemma 2.2 Let U is a positive operator on Hilbert space H. Then for any f H, Uf = 0 if and only if Uf,f =0. Proof. Clearly, Uf = 0 implies Uf,f =0. Conversely, If Uf,f =0, then we have Uf,f = U 1 2 f,u 1 2 f = U 1 2 f 2 =0. Thus U 1 2 f = 0, and hence Uf =0. Proposition 2.2 Let {(W i,v i )} be a parseval fusion frame for H. Then for every J I, S J W SJc W is a positive self-adjiont operator on H which satisfies S J W (SJ W )2 = S J W SJc W 0. Proof. Since SW J it follows that SW J SJc W And we compute and SJc W is a positive self-adjiont. are commuting positive, self-adjoint operators, S J W (SJ W )2 = S J W (I H S J W )=SJ W SJc W 0. Theorem 2.4 Let {(W i,v i )} be a parseval fusion frame for H. For each J I, and f H, the following conditions are equavalent.

7 On the equality of fusion frames 1065 (1) vi 2 π W i (f) 2 = vi 2π W i (2) c v2 i π Wi (f) 2 = c v2 i π Wi (3) vi 2π W i (f), c v2 i π W i (f) =0. (4) f,sw J S J c W f =0. (5) SW J f =(SJ W )2 f. (6) SW J SJc W f =0. Proof. (1) (2): This is follow immediately from Theorem 2.2. (3) (4): This is proven by the following equality: vi 2 π Wi (f), vi 2 π Wi (f) = SW J f,s J c W f = f,sw J S J c W f. c (5) (6): This follows from Proposition 2.2. (1) (5): We have (f) 2 (f) 2 = SW J f,f SJ W f,sj W f = (SJ W (SJ W )2 )f,f. By Proposition 2.2, SW J (SW J ) 2 0. Therefore (1)holds if and only if (SW J (SW J ) 2 )f =0, so it follow that (5) holds by Lemma 2.2. (1) (4): By(1), SW J f,f = SJ W f,sj W f. So SJ W SJc W f,f = (SJ W (SW J )2 )f,f =0, which implies (4). (4) (6): By Proposition 2.2, we have that SW J S J c W 0. Hence SW J S J c W f,f = 0 if and only if SW J S J c W f = 0 by Lemma 2.2. References [1] R.J.Duffin and A.C.Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 1952, 72, [2] P.G.Casazza, G.Kutyniok, Frames of Subspaces, Contemp. math., Vol. 345, Amer.Math. Soc., Providence, RI, 2004, [3] P.Gǎvruta, On the duality of fusion frames, J. Math. Anal. Appl., 2007, 333, [4] M.S.Asgari and Amir Khosravi, Frames and bases of subspaces in Hilbert spaces, J. Math. Anal. Appl., 2005, 308(2),

8 1066 Yao Xiyan, Gao Guibao and Mai Ali [5] P.G.Casazza, The art of frame theory, Taiwanese J. Math., 2000,4, [6] C.Heil, D.Walnut, Continuous and discrete wavelet transforms, SIAM Rev., 1989,31, [7] W.Rudin, Functional Analysis, second ed., McGraw-Hill,1991. Received: October, 2008

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