Intern. ::
Frames
Math.
Journal,
Vol. 4, 2003, no. 6, 527
- 537
and Bases in Tensor Product Hilbert
of
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 eventually 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 useful 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 generalize 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. IJxll=l
(2)
529
Frames and Bases in Tensor Product of Hilbert Spaces
The adjoint of a bounded antilinear map T is defined by (3)
for all x E H, y E K.
< T*x,y >=< Ty,x >
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
(4)
IIT*eiIl2
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
L IITuj1l2.
IIITIW =
(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 Then we define the map x 0 y by (x 0 y)(y') =< y, y' > x,
(y' E K).
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,
IIlx0 ylll = Ilxllllyll,
< x0y,x -
Suppose El = {ei}iEI and E2 respectively. Then,
" , , .0y > =< x, x >< y,y >.
= {Uj}jEJ are
(8) (9) (10)
orthonormal bases for Hand K,
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,ihEJnbeframes for HI,'" ,Hn, withframe boundsAI, BI;'" ;An, Bn, respectively. 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, respectively. 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}, respectively. Then by the Parseval identity, for all T E H 0 K we have
L < TUj,xn0Ym(uj) =L < >< =< L < Ym,Uj >Tuj,xn
< T,xn 0Ym > =
>=
j
j
>
TUj,xn
Ym,Uj
L < TUj, < Ym,Uj > Xn >
j
(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)
531
Frames and Bases in Tensor Product of Hilbert Spaces ;
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 > = LL 1< T*ei,Ym > L IIT*eiI12= D L m
m
i
i
m
:::; D
i
12
12
IITUjl12
= DIIITII12,
IITUjl12
= CIIITIW.
j
and similarly,
L m
IITYml12
~
CL i
IIT*ei112 = CL j
Thus
(13) m
Therefore by inequalities (11) and (13), we get
ACIIITIW:::; LLI n
m
< T,xn0Ym >
12:::; BDIIITIW.
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(Yo 0 xo)Tm} is also a frame for H 0 K.
Amir Khosravi and M. S. Asgari
532
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
L 1< T,QTn
AIIIQ*TIW:::;
n
> 12:::;BIIIQ*TIW.
(15)
Now by using (5) and (14) for every j E J, we have AIIQ-111-211ITIW:::;
L 1< T,QTn n
Therefore
{QTn}nEJ is a frame for H @ K.
> 12:::;BIIQI12111TIW
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.
533
Frames and Bases in Tensor Product of Hilbert Spaces
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}nEf be 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
L< = LL
< Q,T > =
i
>=
Q*ei,T*ei
m
L
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
defined by Sx
= 'En
+H
< 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
535
Frames and Bases in Tensor Product of Hilbert Spaces
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}
< x, S-IX >=
= S-lxn
The frame defined by x~
and
2: 1< S-IX, n
Xn > 12.
(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~. n n
(16)
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) (b)
Q 0 T E B(H 0 K) and IIQ0 TII = IIQIIIITII (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.
2.14. Hence
S(T)
= 2: 2: < T, Xn n
m
m
n
m
= 2:(2: j
=
= Lj
TUj 0 Uj, by Proposition
0 Ym > Xn 0 Ym
= 2: 2: < 2: TUj n
Then T
0 Uj, Xn 0 Ym
> Xn
0 Ym
j
j
< Tuj, Xn > Xn) 0 (2: < Uj, Ym > Ym)
n
m
L SI (TUj) 0 S2(Uj) = L SI 0 S2(TUj j
= SI
0 Uj)
j
082(2:
TUj 0 Uj)
j
= SI
082(T). 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~. o
(iii) Obvious. 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 n
and 82(T*) =
2: < T*,T~ > T~. n
Since for every n E J, < T*, T~ >=< T, Tn > and since the map T t---+T* is a linear operator, we conclude that 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 [1] P. G. Casazza, Every frame is a Sum of three (but not two) orthonormal bases and other frame representations, J. Fourier. Anal. Appl. 4 (1999), 727-732. [2] P. G. Casazza, Local thoery of frames and Schauder bases for Hilbert spaces, Illinois J. Math., 43 (1999), 291-306. [3] 1. Daubechies, Ten Lectures on Wavelets. SIAM. Philadelphia (1992). [4] 1. Daubechies, A. Grasmann and Y. Meyer, Painless nonorthogonal expansions, 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
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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,
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F. Unlii, S. Sonmez,
Frames and bases in tensor product of Hilbert spaces
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A. Kurnaz, G. Oturanc, M. E. Kiris,
Quadratic matrix eigenvalue problem for a certain
type of second order boundary-initial value problems M. Darus, N. Tuneski, functions
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On the Fekete-Szeg6
problem
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On strong alpha-logarithmically convex functions
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Holomorphic vector bundles with infinite-dimensional
Riemann surfaces and compact complex spaces M. de la Sen, Pole-placement estimation
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Ebers on compact 583
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..
