G-Frames and Operator-Valued Frames in Hilbert Spaces

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1 International Mathematical Forum, 5, 2010, no. 33, G-Frames and Operator-Valued Frames in Hilbert Spaces Sedighe Hosseini Department of Mathematics, Science and Research Branch Islamic Azad University, Tehran, Iran Amir Khosravi Faculty of Mathematical Sciences and Computer, Tarbiat moallem University 599 Taleghani Ave., Tehran 15618, Iran khosravi Abstract g-frame in Hilbert spaces have been defined by Sun in [14] and operator-valued frames in Hilbert spaces have been defined by Kaftal et al in [10]. We show that tensor product of g-frames is a g-frame, we get some relations between their g-frame operators, and we show that if they are tight, Riesz bases or orthonormal, then so is their tensor product. We study tensor product of operator-valued frames in Hilbert spaces and tensor product of frame representations. Mathematics Subject Classification: 42C15, 46C05, 46L05 Keywords: Frame, frame operator, operator-valued frames, tensor product, g-frames 1 Introduction Frames were first introduced in 1946 by Gabor [7], reintroduced in 1986 by Daubechies, Grossman and Meyer [5], and popularized from then on. Frames have many nice properties which make them very useful in the characterization of function spaces, signal processing, image processing, data compression, sampling theory and many other fields. Many generalization of frames were introduced, e.g., frames of subspaces [1], [2] pseudo frames[12], oblique frames [4], fusion frames [3]and g-frames in [14]. Tensor product is useful in the approximation of multi-variate functions of combination of univariate ones. Khosravi and Asgari in [11] introduced frames in tensor product of Hilbert spaces. g-frame in Hilbert spaces have been defined

2 1598 S. Hosseini, A. Khosravi by Sun in [14] and operator-valued frames in Hilbert spaces have been defined by Kaftal et al in [10]. In this article, we generalize g-frame and operatorvalued frame to tensor product of Hilbert spaces. In section 2 we briefly recall the definitions and basic properties of Hilbert spaces. In section 3 subsection 3.1 we show also that tensor product of complete, Riesz basis and orthonormal g-frames for Hilbert spaces H and F, present complete, Riesz basis and orthonormal g-frames for H F. In subsection 3.2, we briefly recall the definitions and some properties of operator valued frame in Hilbert spaces which were introduced in [10], and we get some results. We show that tensor product of operator-valued frames for Hilbert spaces H and F present operator-valued frame for H F and we show that tensor product of their analysis operators is an analysis operator. In subsection 3.3, we study the frame representation and we show that tensor product of frame generators is a frame generator. Throughout this paper, N and C will denote the set of natural numbers and the set of complex numbers, respectively. I and J and I, js will be countable index sets. 2 Preliminaries In this section we briefly recall the definition and basic properties of Hilbert spaces. Throughout this paper, H, F are two Hilbert spaces and {V i : V i F, i I}is a sequence of Hilbert spaces, B(H, F) is the collection of all bounded linear operators from Hinto F. Definition 2.1 A family {Λ i B(H, V i ),i I} is said to be a g-frame for H with respect to {V i,i I}, if there are real constants 0 <A B<, such that for all x H, A x 2 i I Λ ix 2 B x 2 (1). The optimal constants (i.e. maximal for A and minimal for B) are called g-frame bounds. The g-frame {Λ i B(H, V i ),i I}is said to be a tight g- frame if A = B, and said to be parseval g-frame if A = B =1. The family {Λ i B(H, V i ),i I} is said to be a g-bessel sequence for H with respect to {V i,i I}, if the right-hand side inequality of (1) holds. Definition 2.2 Let H be a Hilbert space and (V i ) i I is a family of Hilbert spaces and let Λ i B(H, V i ) for each i I. Then (i) If {x :Λ i x =0i I} = {0}then we say that {Λ i } i I is g-complete for Hilbert space H; (ii) If {Λ i } i I is g-complete and there are positive constants Aand B such that for any finite subset I 1 I and y i V i,i I 1, A i I 1 y i 2 i I 1 Λ i y i 2 B i I 1 y i 2, then we say that {Λ i } i I is a g-riesz basis for H with respect to {V i } i I ;

3 G-Frames and operator-valued frames in Hilbert spaces 1599 (iii) We say {Λ i } i I is a g-orthonormal basis for Hwith respect to {V i } i I if it satisfies the following conditions. < Λ i1y i1, Λ i2y i2 >= δ i1,i2 < y i1,y i2 >, i 1,i 2 I,y i1 V i1,y i2 V i2 and i I Λ ix 2 = x 2 x H. Definition 2.3 Let H and H 0 be Hilbert spaces. B(H, H 0 ) is the collection of bounded operators from H to H 0. A collection {A j } j J of operators A j B(H, H 0 ) indexed by a countable set J is called an operator-valued frame on H with range in H 0, if the series S A = j J A j A j converges in the strong operator topology to a positive bounded invertible operator S A. The frame bounds a and b are the largest number a>0 and the smallest number b>0 for which ai S A bi. If a = b, i.e., S A = ai, then the frame is called tight; if S A = I, the frame is called Parseval. Definition 2.4 Given a Hilbert space H 0 and an index set J, define the partial isometries L j : H 0 l(j) H 0 by L j (h) =e j h, where {e j } is the standard basis of l 2 (J). Then L jl i = I 0 if i=j, L jl i =0if i j and j L jl j = I I 0 where I denotes the identity operator on l 2 (J) and I 0 denotes the identity operator on H 0. For operator-valued frame {A j } j J on Hilbert space H with range in H 0, the series j L ja j converges in the strong operator topology to an operator θ A B(H, l 2 (J) H 0 ) and S A = θa θ A. Operator θ A B(H, l 2 (J) H 0 ) is called the analysis operator and the projection P A = θ A S 1 A θ A B(l2 (J) H 0 ) is called the frame projection of {A j } j J. Now we recall some definitions from [9] Tensor product K of two Hilbert spacesh and F is characterized (Up to isomorphism) by the existence of a bilinear mapping p, from the Cartesian product H F into K, with following property:each suitable bilinear mapping L from H F into a Hilbert space K has a unique factorization L = TP, with T a bounded linear operator from K into K. Proposition [9] asserts, in effect, that the only finite families of simple tensors that have sum zero are those that are forced to have zero sum by the bilinearity of the mapping p :(x, y) x y. From this, K 0 can be identified with the algebraic tensor product of H and F, which was defined, traditionally, as the quotient of the linear space of all formal finite sums of simple tensors by the subspace consisting of those finite sums that must vanish if p is to be bilinear. K 0 has the universal property that characterizes the algebraic tensor product. We can identify K with the completion of its everywhere-dense subspace K 0. Accordingly, the Hilbert tensor product H F can be viewed as the completion of the algebraic tensor product K 0, relative to the unique inner product on K 0 that satisfies <x 1 y 1,x 2 y 2 >=< x 1,x 2 >< y 1,y 2 > (x 1,x 2 H, y 1,y 2 F ). For more details see [9].

4 1600 S. Hosseini, A. Khosravi 3 Frames in Hilbert spaces Throughout this section V,W,H and F are Hilbert spaces. {V i : V i V } i I and {W j : W j W } j J are family of Hilbert spaces. B(H, F) is the collection of all bounded linear operators from H into F. 3.1 g-frames in Hilbert spaces Lemma 3.1 Let {Λ j } j J be a g-frame (g-complete, g-riesz basis) in Hilbert space H with respect to {V j } j J with g-frame operator S, let F be a Hilbert space and Q B(H, F) be invertible. Then {Λ j Q } j J is a g-frame(g-complete, g- Riesz basis) for F with respect to {V j } j J with g-frame operator QSQ. Moreover if Q is unitary and {Λ j } j J is a g-orthonormal basis, then {Λ j Q } j J is a g-orthonormal basis. proof. Since Q is invertible and Q B(H, F), then it has an invertible adjoint Q B(F, H). Then by Theorem 3.2 of [12], {Λ j Q } j J is a g-frame with g-frame operator QSQ. If {Λ j } j J is g-complete, then {f :Λ j Q f = 0,j J} = {f : Q f =0} = {0}, because Q is one-to-one. Hence {Λ j Q } j J is g-complete. If {Λ j } j J is a g-riesz basis with bounds A and B, then for every finite subset J 1 of J we have A j J 1 g j 2 j J 1 Λ j g j 2 B j J 1 g j 2.So A Q 1 2 j J 1 g j 2 1 Q 1 2 j J 1 Λ jg j 2 j J 1 QΛ jg j 2 Q 2 j J 1 Λ jg j 2 B Q 2 j J 1 g j 2. For the moreover statement, for each j 1,j 2 J and x j1 V j1,x j2 V j2 we have <QΛ j1x j1,qλ j2x j2 >=< QQ Λ j1x j1, Λ j2x j2 > =< Λ j1 x j1, Λ j2 x j2 >= δ j1j2 <x j1,x j2 > and for each f H, j J Λ jq f 2 = Q f 2 = f 2. Theorem 3.2 Let H, F, W and V be Hilbert spaces and {V j } j J V, {W i } i I W be sequences of closed Hilbert subspaces of V and W, respectively. Let {Λ j } j J be a g-orthonormal basis (resp. g-complete, g-riesz basis) for H with respect to {V j } j J and {Γ i } i I be a g-orthonormal basis(resp. g-complete, g-riesz basis) for F with respect to {W i } i I. Then {Λ j Γ i } i I,j J is a g- orthonormal basis (resp. g-complete, g-riesz basis) for Hilbert space H F with respect to {V j W i } i I,j J.

5 G-Frames and operator-valued frames in Hilbert spaces 1601 proof. We know that (Λ j Γ i ) =(Λ j Γ i ). Let y j1 V j1,y j2 V j2 and x i1 W i1,x i2 W i2. Then we have < Λ j1y j1, Λ j2y j2 >= δ j1,j2 <y j1,y j2 > and for each y H,, j < Λ jy, Λ j y>=< y,y>.also < Γ i1x i1, Γ i2x i2 >= δ i1,i2 <x i1,x i2 > and for each x F,, i < Γ ix, Γ i x>=< x,x>. Therefore < Λ j1 Γ i1 (y j1 x i1 ), Λ j2 Γ i2 (y j2 x i2 ) > =< Λ j1 y j1 Γ i1 x i1, Λ j2 y j2 Γ i2 x i2 > =< Λ j1 y j1, Λ j2 y j2 > < Γ i1 x i1, Γ i2 x i2 > = δ j1,j2 δ i1,i2 <y j1,y j2 > <x i1,x i2 > = δ j1,j2 δ i1,i2 <y j1 x i1,y j2 x i2 >. i,j < Λ j Γ i (x y), Λ j Γ i (x y) = i,j < Λ jx Γ i y, Λ j x Γ i y> = i,j < Λ jx, Λ j x> < Γ i y, Γ i y>= j < Λ jx, Λ j x> i < Γ iy, Γ i y> =< x,x> <y,y>=< x y, x y>. Hence relations (iii) hold and therefore {Λ j Γ i } j J,i I is a g-orthonormal basis for H F. If {Λ j } j J and {Γ i } i I are g-complete, then we have {x :Λ j x =0} = {0} and {y :Γ i y =0} = {0}. Since Λ j x Γ i y = Λ j x Γ i y = 0 then Λ j x =0 or Γ i y =0so {x y :(Λ j Γ i )(x y) =0} = {x y :Λ j x Γ i y =0} = {x :Λ j x =0} {y :Γ i y =0} = {0}. Hence {Λ j Γ i } i I,j J is g-complete. If {Λ j } j J and {Γ i } i I are g-riesz bases, then there exist constants 0 < A B< such that for each finite subset J 1 of J and g j V j A j J 1 <g j,g j > j J 1 < Λ j g j, Λ j g j > B j J 1 <g j,g j >. Also there exist constants 0 <C D< such that for each finite subset I 1 of I and f i W i we have C i I 1 <f i,f i > i I 1 < Γ i f i, Γ i f i > D i I 1 <f i,f i >.

6 1602 S. Hosseini, A. Khosravi Therefore for each finite subset J 1 of J and I 1 of I and every g j V j and f i W i we have AC i,j <g j f i,g j f i > = AC i,j <g j,g j > <f i,f i > C< j Λ j g j, j Λ j g j > i <f i,f i > < j Λ j g j, j Λ j g j > < i Γ i f i, i Γ i f i > B j <g j,g j > D i <f i,f i > = BD i,j <g j f i,g j f i >. 3.2 operator-valued frames in Hilbert spaces We show that tensor product of operator valued frame is an operator valued frame. Theorem 3.3 Let H, F, H 0 and F 0 be Hilbert spaces. Let{A j } j J be an operator-valued frame on H with range in H 0 and {B i } i I be an operator-valued frame on F with range in F 0. Then {A j B i } j J,i I is an operator-valued frame on H F with range in H 0 F 0. In particular, if {A j } j J and {B i } i I are tight or parseval frames, then so is {A j B i } j J,i I. Proof. There is a positive bounded invertible operator S A B(H) such that j A j A J converges in strong operator topology to S A, and there is a positive bounded invertible operator S B B(F ) such that i B i B i converges in strong operator topology to S B. Let x H, y F. Then (A j B i ) (A j B i )(x y) = (A ja j Bi B i )(x y) j J 1,i I 1 j J 1,i I 1 = j J 1 A j A jx i I 1 B i B iy which converges to S A (x) S B (y) and we have the result. From these it follows that for all z = k=n k=1 x k y k in H F, (A j B i ) (A j B i )(z) (S A S B )(z) 0 j J 1,i I 1 This completes the proof. Theorem 3.4 Let {A j } j J be an operator-valued frame in B(H, H 0 ) with analysis operator θ A, {B i } i I be an operator-valued frame in B(F, F 0 ) with analysis operator θ B. Then θ A θ B is the analysis operator of the operator-valued frame {A j B i } j J,i I and P A B = P A P B.

7 G-Frames and operator-valued frames in Hilbert spaces 1603 proof. As we have in definition 2.4, there are partial isometries {L j } j J from H 0 to l 2 (J) H 0, and {K i } i I from F 0 to l 2 (I) F 0 such that L j L s = { I0 (s = j) 0 (s j) j L jl j = I I 0 and θ A = j L ja j. Similarly { Ki K I0 (s = j) s = 0 (s j) i K ik i = I I 0 and θ B = i K ib i. We define partial isometries L j K i : H 0 F 0 l 2 (J I) H 0 F 0,by(L j K i )(h k) =e ji (h k), Where {e ji } is the standard basis of l 2 (J I) θ A B = j i (L j K i )(A j B i )= j i L ja j K i B i = j L ja j i K ib i = θ A θ B. We have P A B = θ A B (S A B ) 1 (θ A B ) =(θ A θ B )(S A S B ) 1 (θ A θ B ) = θ A S 1 A θ A θ BS 1 B θ B = P A P B. Definition 3.5 Let {A j } and {B j } be a pair of operator-valued frames on H. Then {B j } is called a dual of {A j } if θ B θ A = I, the operator-valued frame {A j S 1 A } is called canonical dual frame of {A j}. Lemma 3.6 Let {A j : j J} and {B j : j J} be a pair of dual operatorvalued frames in B(H, H 0 ) and {a k }, {b k } be a pair of dual frames for H 0,respectively. Then {A ja k } and {B j b k } are pair of dual frames for H. Moreover, suppose that {A j : j J} and {B j : j J} are canonical dual operator-valued frames, {a k }, {b k } are canonical dual frames, and that {a k } is a tight frame with frame bound A. Then {A ja k } and {B j b k } are canonical dual frames. proof. {A j : j J} is an operator-valued frame, so j J A j A j = S A with the frame bounds a and b for which ai S A bi. For every η H, we have C<A j η, A j η> < A j η, a k > 2 D<A j η, A j η>. k Consequently C j <A j η, A j η> j < A j η, a k > 2 D j k <A j η, A j η>.

8 1604 S. Hosseini, A. Khosravi So C<S A η, η > j k <η,a ja k > 2 D<S A η, η >, therefore ac < η, η > C<S A η, η > j k <η,a j a k > 2 bd < η, η >. Hence {A ja k } is a frame for H with frame bounds ac andbd. Similarly we can get that {Bj b k} is a frame for H. On the other hand, for any η H, we have j k <η,a ja k >Bj b k = j B j k <A jη, a k >b k = j B j A j η = θ θη = Iη = η Similarly we can get that j k <η,b j b k >A j a k = η. Hence {A j a k} j,k and {Bj b k} j,k are dual frames for H. Now we assume that {A j } and {B j } are canonical dual operator-valued frames and {a k } is a tight frame with frame bound A. Then b k = A 1 a k. Let S A,a be frame operator associated with {A j a k} j,k. Then we have S A,a η = j <η,a j a k >A j a k = j k A j <A j η, a k >a k k = A j A ja j η = AS A η Hence S 1 A,a A j a k = 1 A S 1 A A j a k = B j b k. This completes the proof. Lemma 3.7 Let{A j } and {B j } be a pair of dual operator-valued frames on H, and let {C i }, {D i }be a pair of dual operator-valued frames on F. Then {A j C i } and {B j D i } are dual operator-valued frames on {H F }. proof. Since {A j C i } is an operator-valued frame on {H F } with θ A C = θ A θ C and {B j D i } is an operator-valued on {H F } with θb D = θb θ D, then we have θ B D θ A C =(θb θ D )(θ A θ C )=θb θ A θd θ C=I H I F. This completes the proof. Definition 3.8 An operator-valued frame {A j : j J} for which P A = I I 0 is called Riesz frame and an operator-valued frame that is both parseval and Riesz, i.e., such that θa θ A = I and θ A θa = I I 0, is called an orthonormal frame. Lemma 3.9 Let {A j } j J be a Riesz frame(orthonormal frame) in B(H, H 0 ) and {B i } i I be a Riesz frame(orthonormal frame) in B(F, F 0 ). Then {A j B i } j J,i I is a Riesz frame(orthonormal frame) on H F with range in H 0 F 0 proof. Since {A j } j J, {B i } i I are Riesz frames, we have P A = I H I H0 and P B = I F I F0 so P A B = P A P B = I H F I H0 F 0

9 G-Frames and operator-valued frames in Hilbert spaces frame representation First we recall some definitions and properties of representations from [10]. Let G be a discrete group, not necessarily countable and let λ be the left regular representation of G (resp. ρ the right regular representation of G). Denote by L(G) B(l 2 (G)), (resp. R(G)) the Von Neumann algebra generated by the unitaries {λ g } g G, (resp., {ρ g } g G )It is well known that both L(G) = R(G) and R(G) = L(G) are finite Von Neumann algebras that share a faithful trace vector χ e, where {χ g } g G is the standard basis of l 2 (G). Let H 0 be a Hilbert space and I 0 be the identity of B(H 0 ). We call λ id : g G λ g I 0 the left regular representation of G with multiplicity H 0. Definition 3.10 Let (G, π, H) be a unitary representation of the discrete group G on the Hilbert space H. Then an operator A B(H, H 0 ) is called a frame generator or (resp. a parseval frame generator) with range in H 0 for the representation if {A g := Aπ g 1} g G is an operator-valued frame. (resp. a parseval operator-valued frame). We show that tensor product of representations is a representation. Theorem 3.11 Let (G 1,π 1,H) and (G 2,π 2,F) be unitary representations of discrete groups G 1,G 2 on Hilbert spaces H, F, with frame generators A and B, respectively. Then π 1 π 2 : G = G 1 G 2 B(H F ) is a unitary representation with frame generator A B. If θ A and θ B are the analysis operators for frame generators A and B, respectively, then θ A θ B is the analysis operator for A B. proof. G = G 1 G 2, the direct sum of G 1 and G 2, is a discrete group. We consider the representation π 1 π 2 : G B(H F ), defined by (π 1 π 2 )(g, h) =π 1 g π 2 h (g, h) G Since {A g = Aπ 1g 1 : g G 1 } is an operator-valued frame for H and {B h = Bπ 2h 1 : h G 2 } is an operator valued frame for F, by Theorem 3.3. and the definition of π 1 π 2, {(A B)(π 1g 1 π 2h 1) (g, h) G} is an operator valued frame for H F. Moreover, if θ A and θ B are the analysis operators on H and F for frame generators A andb, respectively, then by Theorem 3.4. θ A θ B is the analysis operator on H F for frame generator A B. References [1] Asgari M. S., and Khosravi A., Frames and bases of subspaces in Hilbert spaces, J. Math. Anal. Appl. 308 (2) (2005) [2] Casazza P. G., and Kutyniok G., Frames of subspaces, in Wavelets, Frames, and Operator Theory. College Park, M.D., 2003 Contemporary Math. 345 (Amer. Math. Soc. Providence, RI, 2004),

10 1606 S. Hosseini, A. Khosravi [3] Casazza P. G., and Kutyniok G., and Li S., Fusion frames and distributed processing, Appl. Comp. Harm. Anal. 25, (2008) [4] Christensen O., and Elder, Y. C, Oblique dual frames and shift-invariant spaces, Appl. Comp. Harm. Anal. 17 (2004) [5] Daubechies, I., and Grossman, A., and Meyer Y., Painless nonorthogonal expansions, J. Math. Phys. 27 (1986) [6] Fornasier, M., Decompositions of Hilbert spaces, in: Local constructions of global frames, Proc. Int. Conf. on constructive function theory, Varna, 2002, B. Bojanov Ed., DARBA (Sofia) (2003) [7] Gabor, D., Theory of communications, J. Inst. Electer. Eng. (London) 93 (3) (1946) [8] Heil, C., E., and Walnut, D. F., Continuous and discrete wavelet transforms, SIAM Review 31 (1989) [9] Kadison, R. V., Ringrose, J.R., Fundamentals of the theory of operator algebras, Vol. I, Pure and applied mathematics (Academic Press) (1983). [10] Kaftal, V., Larson, D. and Zhang, S., Operator-valued frames, preprint. [11] Khosravi, A. and Asgari, M. S., Frames and bases in tensor product of Hilbert spaces, Int. J. Math. 4 (6) (2003) [12] Khosravi, A. and Khosravi, B., Fusion frames and g-frames in Hilbert C*-modules, Int. J. Wavelet, Multiresolution and Information Processing, 6 (3), 2008, [13] Li, S., and Ogawa H., Pseudoframes for subspaces wiyh applications, J. Fourier Anal. Appl. 10 (2004) [14] Sun, W., g-frames and g-riesz bases, J. Math. Anal. Appl., 322 (2006) Received: January, 2010