MULTIPLIERS OF GENERALIZED FRAMES IN HILBERT SPACES. Communicated by Heydar Radjavi. 1. Introduction

Bulletin of the Iranian Mathematical Society Vol. 37 No. 1 (2011), pp 63-80. MULTIPLIERS OF GENERALIZED FRAMES IN HILBERT SPACES A. RAHIMI Communicated by Heydar Radjavi Abstract. In this paper, we introduce the concept of g-bessel multipliers which generalizes Bessel multipliers for g-bessel sequences and we study the properties of g-bessel multipliers when the symbol m l 1, l p, l. Also, we review the behavior of these operators when the parameters are changing. Furthermore, we show that equivalent g-frames have equivalent multipliers and conversely. Finally, we specialize the results to fusion frames. 1. Introduction Frames have been introduced by Duffin and Schaeffer in [11], in connection with non-harmonic Fourier series, but they have attracted more attention since the beginning of wavelet theory. Frames have many nice properties which mae them very useful in the characterization of function spaces, signal processing and many other fields. We refer to [9, 12, 20] for an introduction to the frame theory and its applications. G-frames, introduced by W. Sun in [29], are a natural generalization of frames which cover many other extensions of frames, e.g. bounded quasi-projectors [16, 17], pseudo-frames [21], frame of subspaces or fusion frames [7], outer frames [1], oblique frames [10, 14], continuous frames MSC(2010): Primary: 42C15; Secondary: 41A58, 47A58. Keywords: Frames, g-frame, g-bessel, g-riesz base, g-orthonormal bases. Received: 10 August 2009, Accepted: 2 November 2009. c 2011 Iranian Mathematical Society. 63

64 Rahimi [9] and a class of time-frequency localization operators [13]. Also it was shown that g-frames are equivalent to stable spaces splitings studied in [24]. All of these concepts are proved to be useful in many applications. The perturbation of g-frames and some other properties of this family of operator have been studied in [23, 30]. Bessel multipliers are investigated by Peter Balazs [2, 3, 4] for Hilbert spaces. For Bessel sequences, the investigation of the operator M = m f, ψ ϕ, where the analysis coefficients f, ψ are multiplied by a fix symbol (m ) before resynthesis ( with ϕ ), is very natural and useful and there are numerous applications of this ind of operators. Bessel multipliers of p-bessel sequences in Banach spaces are introduced in [27]. In this paper, by using operator theory tools, we investigate multipliers for g-bessel sequences. We show that when the symbol m c 0, then this operator is compact and when m l 1, l 2, l p it is a trace class, Hilbert-Schmidt and Schatten p-class operator, respectively. Also, we show that equivalent g-frames have equivalent multipliers and conversely. Finally, we specialize the results to fusion frames. 2. G-Frames Through this paper, H and K are Hilbert spaces and { H i : i I} is a sequence of Hilbert spaces, where I is a subset of Z. L(H, H i ) is the collection of all bounded linear operators from H to H i. Note that for any sequence { H i : i I}, we can assume that there exits a Hilbert space K such that for all i I, H i K ( for example K = i I H i). Definition 2.1. A sequence { Λ i L(H, H i ) : i I } is called generalized frame, or simply a g-frame, for H with respect to {H i : i I} if there exist constants A and B, 0 < A B <, such that (2.1) A f 2 i I Λ i f 2 B f 2, f H. The numbers A and B are called g-frame bounds. {Λ i : i I} is called tight g-frame if A = B and Parseval g-frame if A = B = 1. If in (2.1) the second inequality holds, then the sequence is named g-bessel sequence. { Λ i L(H, H i ) : i I } is called a g-frame sequence if it is a g-frame for span{λ i (H i)} i I.

Multipliers of generalized frames in Hilbert spaces 65 It is easy to see that, if {f i } i I is a frame for H with bounds A and B, then by putting H i = C and Λ i ( ) =, f i, the family {Λ i : i I} is a g-frame for H with bounds A and B. Example 2.1. Let H i = C and let Λ i : H C be defined by Λ i f = 1 i f, for any f H. Then, i Λ if 2 = i 1 f 2 = π2 i 2 6 f 2, and so {Λ i : i I} is a tight g-frame for H. Similar to g-frames, the means g-complete, g-orthonormal bases and g-riesz bases can be defined as follows. Definition 2.2. The family { Λ i L(H, H i ) : i I } is called g- complete if {f : Λ i f = 0, i I } = {0} and it is called g-orthonormal basis for H if it satisfies Λ i g i, Λ jg j = δ i,j g i, g j, i, j I, g i H i, g j H j, Λ i f 2 = f 2, f H. i I { Λ i L(H, H i ) : i I } is called g-riesz basis if it is g-complete and there are positive constants A and B such that for any finite subset J I and f i H i, i J, A i J f i 2 i J Λ i f i 2 B i J f i 2. It is clear that if {e i } i I is an orthonormal basis ( or ONB) for H, then {Λ i ( ) =, e i, i I} is a g ONB. Let ( H i = )l 2 i I with the inner product defined by { {f i } i I f i H i, i I and } f i 2 < +, i I {f i }, {g i } = i I f i, g i. ( It is clear that i I i) H is a Hilbert space. Let E i = (e i ) K i be l 2 an orthonormal basis for H i. Define F i, = (0, 0,..., e i, 0,...). Then, }{{} ( i times F = ((F i, ) Ki ) i N is an ONB of i I i) H l 2.

66 Rahimi Proposition 2.3. [23] Λ = { Λ i L(H, H i ) : i I } is a g-bessel sequence for H with bound B if and only if the operator ( T Λ : H i H, )l 2 i I defined by T Λ ({f i } i I ) = Λ i (f i ), i I is a well-defined and bounded operator with T Λ B. Furthermore, ( ) TΛ : H Hi T Λ(f) = {Λ i f} i I. l 2, If Λ = { Λ i L(H, H i ) : i I } is a g-frame, then the operators T Λ and TΛ in Proposition 2.3 are called synthesis operator and analysis operator of Λ = { Λ i L(H, H i ) : i I }, respectively. Definition 2.4. Let Λ = { Λ i L(H, H i ) : i I } be a g-frame for H. The operator S Λ : H H, S Λ = T Λ TΛ is called the g-frame operator of Λ = { Λ i L(H, H i ) : i I }. For any f H, we have S Λ f = i I Λ i Λ i f, AI S Λ BI. It is nown [23] that S Λ is a bounded, positive and invertible operator and every f H has an expansion f = i Λ i Λ is 1 Λ f. One of the most important advantages of g-frames is a resolution of identity i Λ i Λ is 1 Λ = I. Proposition 2.5. [23] Λ = { Λ i L(H, H i ) : i I } is a g-frame for H if and only if the synthesis operator T Λ is well-defined, bounded and onto.

Multipliers of generalized frames in Hilbert spaces 67 Definition 2.6. If Λ = { Λ i L(H, H i ) : i I } and Θ = { Θ i L(H, H i ) : i I } are g-bessel sequences for H and T Λ TΘ = Id, then (Λ, Θ) is called a dual pair. For example, (Λ, S 1 Λ) is a dual pair. Lemma 2.7. [23] If Λ = { Λ i L(H, H i ) : i I } and Θ = { Θ i L(H, H i ) : i I } are g-bessel sequences for H and (Λ, Θ) is a dual pair, then both of Λ and Θ are g-frames for H. 3. Multipliers of Frames Gabor multipliers [13, 15], Gabor filters [22] and some other applications of frames led Peter Balazs to introduce Bessel and frame multipliers for abstract Hilbert spaces H 1 and H 2. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators [2, 3, 4] Definition 3.1. Let H 1 and H 2 be Hilbert spaces, {ψ } H 1 and {φ } H 2 be Bessel sequences. Fix m l. The operator M m,(φ ),(ψ ) : H 1 H 2, defined by M m,(φ ),(ψ )(f) = m f, ψ φ f H, is called the Bessel multiplier for the Bessel sequences {ψ } and {φ }. The sequence m is called the symbol of M. For frames, we will call the resulting Bessel multiplier a frame multiplier and for Riesz sequence a Riesz multiplier. The interested reader can find the properties of this operator in [2, 3, 4]. 4. Multipliers of G-Frames In this section, the concept of multiplier operators for g-bessel sequences will be introduced and some of their properties will be shown. Proposition 4.1. If Λ and Θ are g-bessel sequences with bounds B Λ and B Θ and m l, then the operator M = M m,λ,θ : H H, defined by M(f) = i m iλ i Θ if, for all f H, is well defined.

68 Rahimi Proof. For any f, g H, i m i Λ i Θ i f, g = i m i Θ i f, Λ i g and i m i Θ i f, Λ i g m Θ i f, Λ i g i m ( i Θ i f 2 ) 1 2 ( Λ i g 2 ) 1 2 m BΘ f B Λ g. i Now, we can define multiplier operators for g-bessel sequences. Definition 4.2. Let { H i : i I} be a family of Hilbert spaces. Let Λ = { Λ i L(H, H i ) : i I } and Θ = { Θ i L(H, H i ) : i I } be g-bessel sequences for H with bounds B Λ and B Θ. For m l, the operator M = M m,λ,θ : H H, (4.1) M(f) = i m i Λ i Θ i f, is called the g-bessel multiplier of Λ, Θ with symbol m. It is clear that if m = (m i ) = (1, 1, 1,...) and (Λ, Θ) is a dual pair, then M = Id. Let {λ i } and {ϕ i } be Bessel sequences and m l. Consider the corresponding g-bessel sequences Λ i =, λ i and Θ i =, ϕ i. For any f H, we have M m,λ,θ (f) = M m,(φ ),(λ )(f). The proof of Proposition 4.1 shows that M op = sup{ Mf, f : f = 1} sup m BΘ BΛ f 2 f =1 = m BΘ BΛ.

Multipliers of generalized frames in Hilbert spaces 69 Lemma 4.3. If Λ and Θ are g-bessel sequences and m l, then the adjoint of M m,λ,θ is M m,θ,λ. Proof. For any f, g H, Mf, g = i = i m i Λ i Θ i f, g = i m i Θ i f, Λ i g = i m i Λ i Θ i f, g m i f, Θ i Λ i g, and so M m,λ,θ = i m iθ i Λ i = M m,θ,λ. Lemma 4.4. If Θ = {Θ i } i I is a g-bessel sequence with bound B Θ and m = (m i ) l, then {m i Θ i } i I is a g-bessel sequence with bound m B Θ. Proof. It is easy to show that, for all f H, m i Θ i f 2 m Θ i f 2 m B Θ f 2. i i Lie weighted frames [6], {m i Θ i } i I can be called weighted g-frame (g-bessel). By using the synthesis and the analysis operators of Λ and mθ, respectively, we can write M m,λ,θ f = i So, m i Λ i Θ i f = i (4.2) M m,λ,θ = T Λ T mθ. If we define the diagonal operator ( ) D m : Hi Λ i (m i Θ i )f = T Λ {m i Θ i f} = T Λ T mθf. l 2 ( Hi ) (4.3) D m ((ξ i )) = (m i ξ i ) i I then (4.4) M m,λ,θ = T Λ D m T Θ. We will use both (4.2) and (4.4) in this paper. l 2,

70 Rahimi Proposition 4.5. Let m l, Λ be a g-riesz base and Θ be a g-bessel sequence. The map m M m,λ,θ is injective. Proof. Let M m,λ,θ = 0. Then, for any f H, M m,λ,θ f = 0. Without loss of generality, we can chose f [erθ i ]. Since Λ = { Λ i L(H, H i ) : i I } is a g-riesz base, we have A m i Θ i f 2 Λ i (m i Θ i f) = 0 m i = 0, i I, that is, the map m M m,λ,θ is injective. Proposition 4.6. Let Λ and Θ be g-bessel sequences for H. If m = (m i ) c 0 and (ranθ i ) l, then M m,λ,θ is compact. Proof. Since m = (m i ) c 0, for every ε > 0 there is N N such that m i < ε, for all i > N. Let m N = (m N1, m N2,..., m NN, 0, 0,...). Then, for f H, (TmΘ Tm N Θ)f = {(m i m Ni )Θ i f} l 2 ( m m N Θ i f 2) 1 2 So, T m N Θ T mθ. Since ε B Θ f. m N rantm N Θ = dim ert mn Θ = (dim erθ i ) = ranθ i <, i=1 it follows that Tm N Θ is a finite ran operator. Hence, T mθ is compact. Therefore, M m,λ,θ = T Λ TmΘ M m,λ,θ is compact. i m N i=1 Recall that if T is a compact operator on a separable Hilbert space H, then there exist orthonormal sets {e n } and {σ n } in H such that T x = n λ n x, e n σ n, for x H, where λ n is the n th singular value of T. Given 0 < p <, the Schatten p-class of H [28], denoted by S p, is the space of all compact

Multipliers of generalized frames in Hilbert spaces 71 operators T on H with the singular value sequence {λ n } belonging to l p. It is nown that S p is a Banach space with the norm (4.5) T p = [ n λ n p ] 1 p. S 1 is called the trace class and S 2 is called the Hilbert-Schmidt class. Theorem 1.4.6. in [31] shows that T S p if and only if T p S 1. Moreover, T p p = T p 1. Also, T S p if and only if T p = (T T ) p 2 S 1 if and only if T T S p. Moreover, T p p = T p p = T p p = T p 1 = 2 T T p. 2 It is proved that S p is a two sided -ideal of L(H), that is, a Banach algebra under the norm (4.5) and the finite ran operators are dense in (S p,. p ). Moreover, for T S p, one has T p = T p, T T p and if S L(H), then ST p S T p and T S p S T p. For more information about these operators, see [18, 25, 28, 31]. Analogously, for Hilbert spaces H 1, H 2, H 3, H 4 and for operators A L(H 1, H 2 ), B L(H 4, H 3 ) and C S p (H 3, H 1 ), we have AC S p (H 3, H 2 ) and CB S p (H 4, H 1 ). Theorem 4.7. Let Λ and Θ be g-bessel sequences for H. If m = (m i ) l p and (dimh i ) i I l, then M m,λ,θ is a Schatten p-class operator. Proof. Since S p is a two sided ideal of L(H), it is enough ) to show that the operator D m in (4.4) is in S p. Let F = ((F i, ) Ki be an ONB ( i N of i I i) H. Now, let ( ˆF j ) j N be the rearrangement of F given by: l 2 (1) If 1 j K 1 then ˆF j = F 1,j ; (2) If j > K 1 then ˆF j = F n+1,, where n = max{m N : j > K 1 +... + K m } and = j (K 1 +... + K n ). ( ) So, for every ξ = (ξ i ) i N i I H i where, D m (ξ) = (m i ξ i ) i N = m i ξ, F i, F i, = ˆm j ξ, ˆF j ˆF j, i N K i j N l 2, ( ˆm j ) j N = (m 1, m 1,..., m }{{} 1, m 2, m 2,..., m }{{} 2 K 1 K 2,..., m j, m j,..., m j,...). }{{} K j

72 Rahimi Now, ˆm j p = K i m i p (dimh i ) m p p <, i N j N and therefore, D m S p. The fact that S p is a -ideal of L(H) and (4.4) imply that M m,λ,θ is a Schatten p-class operator. Corollary 4.8. Let Λ and Θ be g-bessel sequences for H. (1) If m = (m i ) l 1 and (dimh i ) i I l, then M m,λ,θ is a traceclass operator. (2) If m = (m i ) l 2 and (dimh i ) i I l, then M m,λ,θ is a Hilbert-Schmit operator. 5. Perturbation of Multipliers Lie Bessel multipliers [2], a g-bessel multiplier M m,λ,θ clearly depends on the parameters m, Λ and Θ. So, it is natural to as: What happens if these items are changed? Proposition 5.1. Let Λ and Θ be g-bessel sequences for H. (1) If m () m in l, then M m,λ,θ M m,λ,θ op 0. (2) If m () m in l p and (dim H i ) l, then M m,λ,θ M m,λ,θ p 0. Proof. The decomposition (4.4) shows that M m,λ,θ M m,λ,θ = M m m,λ,θ = T Λ D m mt Θ. It is easy to observe that (1) M m (),Λ,Θ M m,λ,θ op = T Λ D m mt Θ op and since D m () m op = T Λ op D m () m op T Θ op, sup D m () m (ξ i) (ξ i ) =1 = sup ((m () i m i )(ξ i )) (ξ i ) =1 m () m l 0.

Multipliers of generalized frames in Hilbert spaces 73 So (2) M m,λ,θ M m,λ,θ op 0. M m (),Λ,Θ M m,λ,θ p = T Λ D m mtθ p T Λ op D m () m p TΘ op, and since D m () m p m () m p 0, so M m,λ,θ M m,λ,θ p 0. By an argument lie Proposition 5.1, it can be proved that if Λ Λ strongly, then M m,λ,θ M m,λ,θ. 6. Equivalent G-Frames and their Multipliers In this section we introduce equivalent g-frames and we show that the multipliers of equivalent g-frames are equivalent in the special sense. For more information about equivalent g-frames see [26]. Definition 6.1. Let Λ and Θ be g-frames for H. (1) If Q is an invertible bounded operator and Θ i = Λ i Q, for i I, then we say that Λ and Θ are Q-equivalent. (2) We say they are unitarily equivalent if they are Q-equivalent for a unitary operator Q. (3) If Q is a bounded operator (not necessarily invertible) and Θ i = Λ i Q, then we say that Λ is Q-partial equivalent with Θ. Proposition 6.2. [26] Let Λ and Θ be g-frames with synthesis operators T Λ and T Θ, respectivly. Then, R T Θ R T Λ if and only if there exists a bounded and one to one operator U on H such that Λ i = Θ i U, for any i I. Furthermore, eru = T Λ (R T Λ (R T Θ ) ). Corollary 6.3. Under the assumption of Proposition 6.2, R T Θ = R T Λ if and only if U is a bounded and invertible operator.

74 Rahimi Lemma 6.4. If Λ is a g-bessel sequence with bound B and U : H H is a bounded operator, then ΛU = { Λ i U L(H, H i ) : i I } is a g-bessel sequence. Proof. It is easy to see that for any f H, Λ i Uf 2 B Uf 2 BK 2 f 2. i Let U L(H) and Λ = { Λ i L(H, H i ) : i I } be a g-bessel sequence. Lemma 6.4 shows that ΛU = { Λ i U L(H, H i ) : i I } is a g-bessel sequence, and by a simple calculation we have (6.1) T ΛU = U T Λ and T ΛU = T ΛU. (4.2), (4.4) and (6.1) imply that for bounded operators U, V L(H), and M m,λu,θ = T ΛU D m T Θ = U T Λ D m T Θ = U M m,λ,θ, M m,λ,θv = T Λ D m T ΘV = T Λ D m T ΘV = M m,λ,θ V, M m,λu,θv = T ΛU D m T ΘV = U T Λ D m T ΘV = U M m,λ,θ V. The following theorems may be used as criteria for equivalent g- frames. Theorem 6.5. Let m l, Λ = { Λ i L(H, H i ) : i I }, and Θ = { Θ i L(H, H i ) : i I } and Λ = { Λ i L(H, H i) : i I } be g-bessel sequences. M m,λ,θ = UM m,λ,θ if and only if Λ and Λ are U -equivalent on R T mθ. Proof. M m,λ,θ = UM m,λ,θ T Λ TmΘ = UT Λ T mθ. So, on R TmΘ have T Λ = UT Λ. Therefore, Λ = Λ U and conversely. we Similarly, we can state the following assertion. Theorem 6.6. Let m l, Λ = { Λ i L(H, H i ) : i I }, and Θ = { Θ i L(H, H i ) : i I } and Θ = { Θ i L(H, H i) : i I } be g-bessel sequences. M m,λ,θ = M m,λ,θ V if and only if Θ and Θ are V -equivalent on R TΛ D m.

Multipliers of generalized frames in Hilbert spaces 75 Proof. M m,λ,θ = M m,λ,θ V T Λ D m TΘ = T ΛD m TΘ V. So, on R TΛ D m we have TΘ = T Θ V = T Θ V. Therefore, Θ = Θ V and conversely. 7. Fusion Frames Multipliers Fusion frames (or frames of subspaces) are recent developments of frames that provide a natural mathematical framewor for two-stage (or, more generally, hierarchical) data processing. The notion of a fusion frame was introduced in [8] with the main ideas already presented in [7]. A fusion frame is a frame-lie collection of subspaces in a Hilbert space. In frame theory, a signal is represented by a collection of scalars, which measure the amplitudes of the projections of the signal onto the frame vectors, whereas in fusion frame theory the signal is represented by the projections of the signal onto the fusion frame subspaces. In a two-stage data processing setup, these projections serve as locally processed data, which can be combined to reconstruct a signal of interest. Definition 7.1. Let I be some index set, and {w i } i I be a family of weights, i.e., w i > 0, for all i I. A family of closed subspaces {W i } i I of a Hilbert space H is a fusion frame (or frame of subspaces) with respect to {w i } i I for H, if there exist constants C and D, 0 < C D <, such that (7.1) C f 2 i I w 2 i π Wi (f) 2 D f 2 f H, where π Wi is the projection on the subspace W i. The constants C and D are called the fusion frame bounds. When the upper bound inequality in (7.1) holds, for all f H, {W i } i I is called a fusion Bessel sequence with respect to {w i } i I with Bessel bound D. The following is an example of fusion frames. Example 7.1. Given a function g L 2 (R) and a, b > 0, the Gabor system determined by g and a, b is defined by G(g, a, b) = {E ma T nb g m, n Z}, where E a and T a are unitary operators, defined on L 2 (R), by E a f(x) = e 2πiax f(x), T a f(x) = f(x a).

76 Rahimi G(g, a, b) is unitarily equivalent to G(g, 1 q, 1), where ab = 1 q, and also G(g, 1 q 1 q, 1) = {E 1 j=0 q (mq+1)t na} m,n Z. By letting W j := span m,n Z {E 1 (mq+1)t na} m,n Z, j = 0, 1,...q 1, {W j } q 1 j=0 q is a frame of subspaces (for more details, see [7]). The above example can be related to Gabor multipliers introduced in [19]. It is clear that by putting Λ i = w i π Wi and H i = W i in (2.1), we conclude that fusion frames are a special g-frame, so the results in sections 4 and 5 hold for fusion frames. Definition 7.2. Let W = (W i, w i ) i I and V = (V i, v i ) i I be fusion Bessel sequences with bounds B W and B V, respectively and let m l. The operator M m,w,v : H H defined by (7.2) M m,w,v f := i m i v i w i π W i π Vi f for all f H is called Bessel fusion multiplier of W = (W i, w i ) i I and V = (V i, v i ) i I. Note that we consider π Vi : H H i so its adjoint π V i : H i H is just the inclusion. The results shown in Section 4 can be specialized for fusion frames to get: Theorem 7.3. Let W = (W i, w i ) i I and V = (V i, v i ) i I be fusion Bessel sequences with bounds B W and B V, respectively. (1) If m l, then the operator M m,w,v defined by (7.2) is well defined and M m,w,v op B W B V m. (2) (M m,w,v ) = M m,v,w. (3) If m = (m i ) c 0 and (dimv i ) l, then M m,w,v is compact. (4) If m = (m i ) l p and (dimv i ) l, then M m,w,v is a Schatten p-class operator for all p 1. In particular also for p = 1 (trace class) and p = 2 (Hilbert-Schmidt class). Using the fact that for frame sequences, ψ 2 < if and only if its span is finite-dimensional [5], we can rephrase some items of the above theorem:

Multipliers of generalized frames in Hilbert spaces 77 Corollary 7.4. Let W = (W i, w i ) i I and V = (V i, v i ) i I be fusion Bessel sequences with bounds B W and B V, respectively. Let (φ (i) ) K be frames for V i. (1) If m = (m i ) c 0 and sup i φ(i) <, then M m,w,v is compact. (2) If m = (m i ) l p and sup i φ(i) <, then M m,w,v is a Schatten p-class operator, for all p 1. For fusion frames, we can state (7.2) in the following form. Let {ψ (i) } be a frame for W i and let {φ (i) } be a frame for V i, for all i I. It is easy to show that M m,w,v f = i,,l m i v i w i f, φ (i) l φ (i) l (i) for any f H,. Where { ψ } is a dual of {ψ (i), ψ (i) ψ (i) φ (i) } is a dual } and { of {φ (i) }. (i) In the case that { φ } and {φ (i) } are biorthogonal, we have M m,w,v f = i, m i v i w i f, φ (i) (i) ψ, which corresponds to a frame multiplier in the sense of [2]. As fusion frames were investigated to string together frames of subspaces, it is natural to loo at a combination of fusion frame multipliers and frame multipliers on the related subspaces. Definition 7.5. Let W = (W i, w i ) i I and V = (V i, v i ) i I be fusion frames and let m l. Furthermore, let n (i) = (n (i) ) K l and (ψ (i) ) K and (φ (i) ) K be frames for W i and V i, respectively. Then, we ( can define the combined multiplier M m,w,v [ ( M m,w,v M n (i),ψ (i),φ(i) ) i I ] f := i M n (i),ψ (i),φ(i) ) m i w i v i π Wi M n (i),ψ (i) i I by π Vi f.,φ(i) To shorten the notation, we use M to denote the combined multiplier.

78 Rahimi Let ξ (i) = πw i ψ (i) and η (i) = πv i φ (i), i.e. the extension of the frames of the subspaces to frame sequences of the whole space. Then, we can write Mf = m i w i v i πw i M (i) π Vi f = m i w i v i πw i n (i) π Vi f, φ (i) ψ (i) i i = i, m i w i v i n (i) f, π V i φ (i) π W i ψ (i). So Mf = i, m i n (i) f, v iη (i) w iξ (i). Under this condition, {v i η (i) } i, and {w i ξ (i) } i, are frames for H [7], so this is again a frame multiplier. Therefore, using the above results, we can formulate the following proposition. Proposition 7.6. Let m l and n (i) = (n (i) ) l. Let W = (W i, w i ) i I and V = (V i, v i ) i I be fusion frames with upper bounds B W and B V, respectively. Let (ψ (i) ) K and (φ (i) ) K be frames for W i and V i ; respectively. Let sup i φ(i) <. (1) If either m = (m i ) c 0 and n (i) = (n (i) ) K c 0 or n = (n (i) ) (,i) K I c 0, ( then M m,w,v M n (i),ψ (i),φ(i) ) i I is compact. (2) Let p 1. If either m = (m i ) l p and n = (n (i) ) (,i) K I l (,p) or n = (n (i) ) (,i) K I l (p,p), ( then M m,w,v M n (i),ψ (i),φ(i) ) i I is a Schatten p-class operator. (3) Let p, p 1 and 1 p + 1 p = 1. If m = (m i ) l p and n = ( ) (n (i) ) (,i) K I l (1,p ), then M m,w,v M n (i),ψ (i) is a,φ(i) i I trace class operator.

Multipliers of generalized frames in Hilbert spaces 79 Acnowledgments The author was partly supported by the WWTF project MULAC (Frame Multipliers: Theory and Application in Acoustics, MA07-025). The author would lie to than Peter Balazs, Diana T. Stoeva and the referee(s) for their useful comments and suggestions. References [1] A. Aldroubi, C. Cabrelli and U. Molter, Wavelets on irregular grids with arbitrary dilation matrices and frame atomics for L 2 (R d ), Appl. Comput. Harmon. Anal., 17, (2004) 119-140. [2] P. Balazs, Basic definition and properties of Bessel multipliers, Journal of Mathematical Analysis and Applications, 325(1), (2007) 571-585. [3] P. Balazs, Hilbert-Schmidt Operators and Frames-Classifications, Approximation by Multipliers and Algorithms, International Journal of Wavelets, Multiresolution and Information Processing, 6 (2), (2008) 315-330. [4] P. Balazs, Matrix Representations of Operators Using Frames, Sampling Theory in Signal and Image Processing (STSIP), 7(1), (2008) 39-54. [5] P. Balazs, Frames and Finite Dimensionality: Frame Transformation, Classification and Algorithms, Applied Mathematical Sciences, 2, (2008) 2131-2144. [6] P. Balazs, J-P. Antoine and A. Grybos, Weighted and Controlled frames, International Journal of Wavelets, Multiresolution and Information Processing, 8(1), (2010) 109-132. [7] P. G. Casazza and G. Kutynio, Frames of Subspaces, Wavelets, Frames and Operator Theory. Contemp. Math. Amer. Math. Soc. 345 (2004) 87-113. [8] P. G. Casazza, G. Kutynio, and S. Li, Fusion Frames and Distributed Processing, Appl. Comput. Harmon. Anal., 25 ( 1), (2008)114-132. [9] O. Christensen, Frames and Bases An Introductory course, Birhauser 2008. [10] O. Christensen and Y. C. Eldar, Oblique dual frames and shift-invariant spaces, Appl. Comput. Harmon. Anal.17, (2004) 48-68. [11] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72, (1952) 341-366. [12] I. Daubechies, Ten Lectures on Wavelets, SIAM Philalphia, 1992. [13] M. Dörfler, Gabor analysis for a class of signals called musisc, Ph.D. thesis, University of Vienna, 2003. [14] Y. C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, J. Fourier Anal. Appl. 9, (2003) 77-96. [15] H. G. Feichtinger and K. Nowa, A first survey of Gabor multipliers, Advances in Gabor Analysis, 99-128, Appl. Numer. Harmon. Anal., Birhäuser, Boston, MA, 2003 [16] M. Fornasier, Decomposition of Hilbert spaces: local construction of global frames, Proc. Int. Conf. on Constructive function theory, Varna, B. Bojanov Ed., DARBA, Sofia, (2003) 275-281.

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