Intern. Math. Journal, Vol. 4, 2003, no. 6, 527 - 537

:: Frames and Bases in Tensor Product of Hilbert Spaces

Amir Khosravi

Faculty of Mathematical Sciences and Computer Engineering University For Teacher Education, Taleghani Ave. 599 Tehran 15614, Iran e-mail: [email protected]@yahoo.com

M. S. Asgari

Department of Mathematics, Science and Research Branch Islamic Azad University, Tehran, Iran e-mail: [email protected]

Abstract In this article we develop a theory for frames in tensor product of Hilbert spaces. We show that like bases if YI, ... ,Yn are frames for HI,. .. , Hn, respectively, then

is a frame for HI @... @Hn. Moreover we consider the canonical dual frame in tensor product space. We further obtain a relation between the dual frames in Hilbert spaces, and their tensor product.

;; Mathematics Subject Classification: 41A38, 42C15 and 43A70 Keywords: Frame, bases, Tensor product, Frame operator, Dual frame, Hilbert space. -

1 Introduction

In 1946 Gabor [7]introduced a technique for signal processing which led even- tually to wavelet theory. Later in 1952 Duffin and Schaeffer [5] introduced frame theory for Hilbert spaces. In 1986 Daubechies, Grossmann and Meyer [4] show that Duffin and Schaeffer's definition was an abstraction of Gabor's 528 Amir Khosravi and M. S. Asgari concept. Nowadays frames work as an alternative to orthonormal bases in Hilbert spaces which has many advantages [9]. Since tensor product is use- ful in approximation theory, in this article we consider the frames in tensor product of Hilbert spaces and extend some of the known results about bases to frames. Let H be a separable complex Hilbert space. As usual we denote the set of all bounded linear operators on H by B (H). We use N, Z, IR and C to denote the natural numbers, integers, real numbers and complex numbers, respectively. I, J and every Ji will denote generic countable (or finite) index sets. We will always use El = {ei hE! and E2 = {Uj} jEJ to denote orthonormal bases for Hand K, respectively. A sequence {xn} in a Hilbert space H is called a frame for H, if there exist two constants A, B > 0 such that for all x E H. (1) n The numbers A and B are called the frame bounds. The frame {xn} is called a tight frame if we can choose A = B and a normalized tight frame if A = B = 1. Therefore {xn} is a normalized tight frame if and only if for every x E H, IIxI12= 2:n I < x, Xn> 12.If {Xn} is a normalizedtight frame, then for every x EH, x = 2:n < X,Xn > Xn (at least in the weakly convergent sense). Conversely, if {xn} is a sequence in H such that the equation x = 2:n < x, Xn > Xn holds for every x E H (the convergence can be either in the weakly convergent sense or in the norm convergent sense) then {xn} is a normalized tight frame for H. Obviously every orthonormal basis is a normalized tight frame. Throughout this paper, all of the Hilbert spaces will be separable and complex. For convenience we will denote the inner product of all Hilbert spaces by < .,. >.

2 Frame In Tensor Product

In this section we consider the tensor product of Hilbert spaces and we gener- alize some of the known results about bases to frames. There are several ways of defining the tensor product of Hilbert spaces. Folland in [6], Kadison and Ringrose in [11] have represented the tensor product of Hilbert spaces Hand K as a certain linear space of operators. Since we used their results firstly we state some of the definitions. Let Hand K be Hilbert spaces. Then we consider the set of all bounded antilinear maps from K to H. The operator norm of an antilinear map T is defined as in the linear case:

IITII = sup IITxlI. (2) IJxll=l Frames and Bases in Tensor Product of Hilbert Spaces 529

The adjoint of a bounded antilinear map T is defined by < T*x,y >=< Ty,x > for all x E H, y E K. (3) Note that the map T t---7 T* is linear rather than antilinear. Suppose T is an antilinear map from K into H and El = {eihEI and E2 = {Uj }jEJ are orthonormal bases for Hand K, respectively. Then by the Parseval identity

L IITujll2 = L IIT*eiIl2 (4) j

This shows that L:j IITujl12is independent of the choice of basis E2. Definition 2.1 Let Hand K be Hilbert spaces. Then the tensor product of Hand K is the set H 0 K of all antilinear maps T : K --+ H such that L:j IITUj112< 00 for some, and hence every, orthonormal basis E2 of K. Moreover for every T E H 0 K we set

IIITIW = L IITuj1l2. (5) j

By Theorem 7.12 in [6],H 0 K is a Hilbert space with the norm 111.111 and associated inner product < Q,T >= L < QUj,Tuj >, (6) j

where E2 = {Uj}jEJ is any orthonormal basis of K. Let x E Hand y E K. Then we define the map x 0 y by

(x 0 y)(y') =< y,y' > x, (y' E K). (7) Obviously x 0 y belongs to H 0 K. Let T E H0K. Hx,x' E Hand y,y' E K, then by [6]

IIITIII = IIIT*III, (8) IIlx0 ylll = Ilxllllyll, (9) " , , < x0y,x .0y > =< x, x >< y,y >. (10)

- Suppose El = {ei}iEI and E2 = {Uj}jEJ are orthonormal bases for Hand K, respectively. Then,

El 0 E2 = {ei 0 Uj : i El, j E J} is an orthonormal basis for H 0 K, by Proposition 7.14 in [6]. Now we can generalize Theorem 2.6.4 of [11] and Proposition 7.14 of [6]to frames. 530 Amir Khosravi and M. S. Asgari

Theorem 2.2 Let HI,'" ,Hn beHilbert spaces andYI = {YI,ihEJp'" ,Yn = {Yn,ihEJnbeframesfor HI,'" ,Hn, withframe boundsAI, BI;'" ;An, Bn, re- spectively. Then

is a frame for HI 0 ... 0 Hn with frame bounds AIA2'" An and BIB2'" Bn. In particular, if YI, . .. ,Yn are normalized tight frames, then it is a normalized tight frame.

By using the associativity of tensor product [11, Proposition 2.6.5]and by induction it is enough to prove the theorem for n = 2.

Theorem 2.3 Let {Xn}nEI and {Ym}mEJ be frames for Hand K, respec- tively. Then {xn 0 Ym}nEI,mEJis a frame for H 0 K. Moreover, {xn 0 Ym} is a normalized tight frame if {xn} and {Ym} are.

Proof. Let A, Band C, D be the bounds of the frames {xn} and {Yn}, re- spectively. Then by the Parseval identity, for all T E H 0 K we have

< T,xn 0Ym > = L < TUj,xn0Ym(uj) >= L < TUj, < Ym,Uj > Xn > j j

= L < Ym,Uj > < TUj,xn > j =< L < Ym,Uj >Tuj,xn > (T is an antilinear map) j =< T(L < Ym,Uj > Uj), Xn >=< TYm, Xn > . j

Therefore L:n L:m I < T, Xn 0 Ym > 12 = L:n L:m 1 < TYm, Xn > 12. Since {xn} is a frame for H, it follows that for every m E J2,

n and

(11) m m n m

Moreover, since El is an orthonormal basis for H, then by the Parseval identity

(12) Frames and Bases in Tensor Product of Hilbert Spaces 531 ;

Now by using the fact that {Ym} is a frame for K and by (4), we conclude that

L IITYml12 = LL 1< T*ei,Ym > 12 m m i

=LL 1< T*ei,Ym > 12 i m

:::; D L IIT*eiI12= D L IITUjl12 = DIIITII12, i j and similarly,

L IITYml12 ~ CL IIT*ei112= CL IITUjl12 = CIIITIW. m i j Thus (13) m Therefore by inequalities (11) and (13), we get

ACIIITIW:::;LLI < T,xn0Ym > 12:::; BDIIITIW. n m Thus {Xn 0 Ym} is a frame for H 0 K. <> For the converse we have the following result. Theorem 2.4 Let {Tn}nEJ be a frame for H 0 K. Then for each Xo E H and Yo E K the sequences {TnYo}nEJ and {T~Xo}nEJ are frames for Hand K, respectively. Moreover these are tight frames, if {Tn}nEJ is.

Proof. Let A, B be the frame bounds for {Tn}nEJ. As we saw in the proof of Theorem 2.1, for all x E H we have < x (8)Yo,Tn >=< x, TnYo> . Since {Tn}nEJ is a frame for H 0 K, we have

n Hence

n Therefore {TnYo}nEJ is a frame for H. Similarly, since for all Y E K < Y,T:xo >=< Xo,TnY >=< Xo(8)Y, Tn >, we conclude that {T~XO}nEJ is also a frame for K. <> Corollary 2.5 If {Tn}nEJ is a frame for H 0 K, then for each Xo E Hand Yo E K the sequence {Tn(Yo0 xo)Tm} is also a frame for H 0 K. 532 Amir Khosravi and M. S. Asgari

Proof. In view of Theorems 2.3 and 2.4, the sequence {TnYo@T~xo} is a frame for H @K. On the other hand, since Tn is antilinear, for every Y E K TnYo @T~xo(Y) =< T~xo,Y > TnYo=< Tmy,Xo> TnYo= Tn(Yo@xo)Tm(Y). Hence {Tn(Yo @ xo)Tm} is a frame for H @ K. <> Theorem 2.6 If Q E B(H) is an invertible operator and {Tn}nEJ is a frame in H @ K, then the sequence {QTn}nEJ is also a frame for H @ K.

Proof. Since Q is a bounded invertible operator on H, then for each x E H (14) Let T E H @K. Since {Tn} is a frame for H @K and Q*T EH @K we have

AIIIQ*TIW :::; L 1 < Q*T, Tn > 12 :::; BIIIQ*TIW, n where A, B are frame bounds for {Tn}nEJ. But < Q*T,Tn >=< T, QTn >, therefore AIIIQ*TIW:::;L 1< T,QTn > 12:::;BIIIQ*TIW. (15) n Now by using (5) and (14) for every j E J, we have

AIIQ-111-211ITIW:::; L 1< T,QTn > 12:::;BIIQI12111TIW n Therefore {QTn}nEJ is a frame for H @ K. <> Corollary 2.7 If Q E B(H) is a unitary operator and {Tn}nEJ is a frame in H@K, then the sequence {Rn}nEJ defined by Rn = QTn is also a frame for H@K. Since IIITIII= IIIT*III,then T E H@K if and only ifT* E K@H. Nowwe have the following result. Theorem 2.8 The sequence {Tn}nEJ in the Hilbert space H @K is a frame if and only if {T;}nEJ is a frame for K @H.

Proof. It's enough to note that IIIT*III = IIITIIIand for evrey n, by applying the Parseval identity as in (4), we have < T*,Tn >=< T,T; >. For the converse, it is enough to note that T;* = Tn. <> Corollary 2.9 The sequence {Tn}nEJ in the Hilbert space H@H is a frame if and only if {T;}nEJ is. Corollary 2.10 If Q E B(K) is an invertible operator and {Tn}nEJ is a frame for H @K, then the sequence {TnQ}nEJ is also a frame for H @K. Frames and Bases in Tensor Product of Hilbert Spaces 533

Proof. Since {Tn}nEJ is a frame for H 0 K, {T~}nEJ is a frame for K 0 H, by the above theorem. Moreover, Q E B(K) is an invertible operator, thus Q* E B(K) is an invertible operator. Now by Theorem 2.6, {Q*T~}nEJ is a frame for K 0 H. Therefore, by Theorem 2.8, {TnQ}nEJ is also a frame for H0K. 0

Corollary 2.11 Let {Tn}nEfbe a frame for H 0 K and let Q E B(H) and RE B(K) be invertible operators. Then the sequence {Sn}nEJ defined by Sn = QTnR is a frame for H 0 K. Now we can represent the inner product in H 0 K by normalized tight frames in K.

Theorem 2.12 If {Ym}mEJ is a normalized tight frame for Hilbert space K, then for all Q, T EH 0 K we have

< Q,T >= L < QYm,TYm >. mEJ

Proof. Let T, Q E H 0 K, Let {eihE! be an orthonormal basis for K. Since {Ym} is a normalized tight frame, then for every i E I and T E H 0 K, we have T*ei = 2:m < T*ei, Ym > Ym and IIT*eiI12= 2:m I < T*ei, Ym > 12,Hence

< Q,T > = L < Q*ei,T*ei >= L < Q*ei'L < T*ei,Ym > Ym > m = LL < Q*ei,Ym > < TYm,ei >. i m Now by using the Schwarz inequality, we conclude that the last double series is absolutely convergent, so o m m

Corollary 2.13 1fT E H 0 K and{Yn}nEJp {Zm}mEh be two normalized tight frames for K, then

n m

Proposition 2.14 Let El = {eihE! and E2 = {Uj}jEJ be orthonormal bases for Hand K, respectively,and let T E H 0 K. Then

T = L ei 0 T*ei (pointwise),

T = I:TUj 0 Uj (pointwise). j 534 Amir Khosravi and M. S. Asgari

Proof. By the Parseval identity for every y E K we have

and

Ty = T(L < y,Uj > Uj) = L < y,Uj >Tuj j j = L < uj,y > TUj = LTuj @Uj(Y)' j j

o

Proposition 2.15 Let {Xn}nEh and {Ym}mEh be normalized tight frames for Hand K, respectively, and let T E H @K. Then T = 'En Xn @T*xn = 'Em TYm @ Ym'

Proof. Let Y E K be arbitrary. Then y = 'Em < Y, Ym > Ym and Ty = 'En < Ty, Xn > Xn, in the weak sense. Therefore

n n n and

m m

m m o

3 The Canonical Dual Frame

Let {xn}nEJ be a frame in the Hilbert space H. Then the operator S : H + H defined by Sx = 'En < x, Xn > Xn, (x E H), is called the frame operatorfor {xn}. By Theorem 2.1.3 in [9], {Xn}nEJ is a frame with frame bounds A, B if and only if S is a bounded linear operator with AI ~ S ~ BI where I denotes the identity operator on H. Moreover S is a positive bounded linear invertible Frames and Bases in Tensor Product of Hilbert Spaces 535

operator, and the sequence {S-lxn}nEJ is a frame with frame bounds B-1, A-I for H. Every x E H can be written as

n n Thus S-1 is the frame operator of {S-lxn} and

< x, S-IX >= 2: 1< S-IX, Xn > 12. n The frame defined by x~ = S-lxn (n E J) is called dual frame of {Xn}nEJ in the frame literature. Thus for every x E H, we have

x = 2: < x, x~ > Xn= 2: < x, Xn > x~. (16) n n

Let Hand K be Hilbert spaces. Then by Theorem 7.18 in [6] for all Q, Q' E B(H) and T, T' E B(K) we have (a) Q 0 T E B(H 0 K) and IIQ0 TII = IIQIIIITII (b) (Q 0 T)(x 0 y) = Qx 0 Ty for all x E H, yE K (c) (Q 0 T)(Q' 0 T') = (QQ') 0 (TT') (d) If Q E B(H) and T E B(K) be invertible operators, then Q 0T is an invertible operator and (Q 0 Ttl = Q-l 0 T-l.

Proposition 3.1 Let {Xn}nEh and {Ym}mEh be frames in the Hilbert spaces Hand K respectively, and let SI, S2 and S be the frame operators of {xn}, {Ym} and {xn 0 Ym}, respectively.Then S = SI 0 S2.

Proof. Let T E H 0 K be arbitrary. Then T = Lj TUj 0 Uj, by Proposition 2.14. Hence

S(T) = 2: 2: < T,Xn 0 Ym > Xn 0 Ym n m

= 2: 2: < 2: TUj 0 Uj, Xn 0 Ym > Xn 0 Ym n m j

n m j = 2:(2: < Tuj, Xn > Xn) 0 (2: < Uj,Ym > Ym) j n m

= L SI (TUj) 0 S2(Uj) = L SI 0 S2(TUj 0 Uj) j j

= SI 082(2: TUj 0 Uj) = SI 082(T). j o 536 Amir Khosravi and M. S. Asgari

Proposition 3.2 Let {Xn}nEI and {Ym}mEJ beframes in the Hilbert spaces Hand K, respectively. If 0 =I-,\ E C, then (i) (xn 0 Ym)' = x~ 0 y:n (ii) ('\xn)' = (~)-IX~ (iii) (x~)' = Xn

Proof. (i) Let 81,82 and 8 be the frame operators of {xn}, {Ym} and {xn 0 Ym}, respectively.Then by Proposition 3.1,8 = 81082 and 8-1 = 8:;10821. Thus

(ii) Let T be the frame operator for {'\xn}' Then T-l = 1'\1-28:;1,where 81 is the frame operator of {xn}' Therefore

('\Xn)' = T-l('\xn) = 1,\1-28:;1('\xn) = 1,\1-2,\8:;I(xn) = (~)-IX~. (iii) Obvious. o

Proposition 3.3 Let {Tn}nEJ be a frame for H 0 K. Then (T~)' = (T~)*.

Proof. Let 81 E B(H 0 K) and 82 E B(K 0 H) be the frame operators of {Tn} and {T~}, respectively. Then for every T E H 0 K, we have

81(T) = 2: < T, Tn > Tn and 82(T*)= 2: < T*,T~ > T~. n n Since for every n E J, < T*, T~ >=< T, Tn > and since the map T t---+T* is a linear operator, we concludethat

n n

n

So 82(T*) = (81(T))*. Since 8:;I(Tn) = T~, it follows that Tn = 81(T~). By taking adjoints on both sides, we get T~ = (81n(T~))*= 82((T~)*).Hence

(T~)* = 821(T~) = (T~)' Therefore, (T~)*= (T~)'. o Frames and Bases in Tensor Product of Hilbert Spaces 537

References

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[3] 1. Daubechies, Ten Lectures on Wavelets. SIAM. Philadelphia (1992). [4] 1. Daubechies, A. Grasmann and Y. Meyer, Painless nonorthogonal ex- pansions, J. Math. Phys. 27 (1986), 1271-1283. [5] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (2), (1952), 341-366. [6] G. B. Folland, A Course in abstract harmonic analysis, CRC Press BOCA Raton, Florida (1995).

[7] D.Gabor, Theory of communication, J. Inst. Electr. Eng. London, 93 (Ill) 1946, 429-457.

[8] D. Han and D. Larson, Frames, Bases and Group Representation, Mem. Amer. Math. Soc., 147(2000) 697.

[9] C. E. Heil and D. F. Walnut. Continuous and discrete Wavelet transforms, SIAM. Review 31 (1989), 628-666. [10] J. R. Holub. Pre-frame operators, Besselian frames and near-Riesz bases in Hilbert spaces, Proc. Amer. Math. Soc., 122 (1994), 779-785. [11] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, Vol I, Academic Press, NewYork 1983.

[12] E. A. Light and E. W. Cheney, Approximation theory in tensor product spaces. Lecture notes in Mathematics (1169). Springer-Verlag (1985).

[13] G. J. Murphy. C*-Algebra and operator theory. Academic Press, London (1990).

Received: September 17, 2003 International Mathematical Journal, Vol. 4, no. 6, 2003

Contents

Ming-Ai Jin, Sang-Gu Lee, Cora Neal, Sangwook Ree, A matrix method for the input - output analysis of economic systems 515

A. Khosravi, M. S. Asgari, Frames and bases in tensor product of Hilbert spaces 527

F. Unlii, S. Sonmez, Convoluted Priifer's type I@I 539

A. Kurnaz, G. Oturanc, M. E. Kiris, Quadratic matrix eigenvalue problem for a certain type of second order boundary-initial value problems 549

M. Darus, N. Tuneski, On the Fekete-Szeg6 problem for generalized close-to-convex functions 561

N. Tuneski, M. Darus, On strong alpha-logarithmically convex functions 569

E. Ballico, Direct images of holomorphic vector bundles on pk-projections on infinite - dimensional complex spaces 575

E. Ballico, Holomorphic vector bundles with infinite-dimensional Ebers on compact Riemann surfaces and compact complex spaces 583

M. de la Sen, Pole-placement adaptive control of linear systems with two parametrical estimation schemes 601 ..