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 amir@yahoo.com
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 prod-uct, 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 characteriza-tion of funccharacteriza-tion 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 com-bination of univariate ones . Khosravi and Asgari in [11] introduced frames in tensor product of Hilbert spaces. 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]. In this article, we generalize g-frame and operator-valued 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 spacesH 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 Ij,s 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 {Vi : Vi ⊆
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 {Λi ∈B(H, Vi), i∈I} is said to be a g-frame for H
with respect to {Vi, i∈ I} , if there are real constants 0< A ≤ B <∞, such
that for all x∈H,
A x2≤i∈I Λix2≤B x2 (1).
The optimal constants (i.e. maximal for A and minimal for B) are called g-frame bounds. The g-frame {Λi ∈ B(H, Vi), 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
{Λi ∈B(H, Vi), i ∈I} is said to be a g-Bessel sequence for H with respect to
{Vi, i∈I} , if the right-hand side inequality of (1) holds.
Definition 2.2 Let H be a Hilbert space and (Vi)i∈I is a family of Hilbert
spaces and let Λi ∈B(H, Vi) for each i∈I. Then
(i) If {x : Λix = 0 i ∈ I} = {0}then we say that {Λi}i∈I is g-complete for
Hilbert space H;
(ii) If {Λi}i∈I is g-complete and there are positive constants Aand B such that
for any finite subset I1 ⊂I and yi ∈Vi, i∈I1,
Ai∈I
1 yi 2≤
i∈I1Λ∗iyi 2≤B
i∈I1 yi 2, then we say that {Λi}i∈I is
(iii) We say {Λi}i∈I is a g-orthonormal basis for Hwith respect to {Vi}i∈I if it
satisfies the following conditions.
< Λ∗i1yi1,Λ∗i2yi2 >= δi1,i2 < yi1, yi2 >,∀i1, i2 ∈ I, yi1 ∈ Vi1, yi2 ∈ Vi2 and
i∈I Λix2=x2 ∀x ∈H.
Definition 2.3 Let H and H0 be Hilbert spaces. B(H, H0) is the collec-tion of bounded operators from H to H0. A collection {Aj}j∈J of operators
Aj ∈ B(H, H0) indexed by a countable set J is called an operator-valued
frame on H with range in H0, if the series SA = j∈JA∗jAj converges in
the strong operator topology to a positive bounded invertible operator SA. The
frame bounds a and b are the largest number a > 0 and the smallest number
b > 0 for which aI ≤ SA ≤ bI. If a = b, i.e., SA = aI , then the frame is
called tight; if SA=I, the frame is called Parseval.
Definition 2.4 Given a Hilbert spaceH0 and an index setJ, define the partial isometries
Lj : H0 → l(J)⊗H0 by Lj(h) = ej ⊗h, where {ej} is the standard basis of
l2(J). Then L∗jLi =I0 if i=j, L∗jLi = 0 if i=j and jLjL∗j =I⊗I0 where I
denotes the identity operator on l2(J) and I0 denotes the identity operator on
H0. For operator-valued frame {Aj}j∈J on Hilbert space H with range in H0,
the series jLjAj converges in the strong operator topology to an operator
θA ∈ B(H, l2(J)⊗H0) and SA = θ∗AθA. Operator θA ∈ B(H, l2(J)⊗H0) is
called the analysis operator and the projection PA=θASA−1θA∗ ∈B(l2(J)⊗H0)
is called the frame projection of {Aj}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 productH×F into K, with following property:each suitable bilinear mapping L from H ×F into a Hilbert space K has a unique factorization L = T P, with T a bounded linear operator from K into K. Proposition 2.6.6 [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 , K0 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. K0 has the universal property that characterizes the algebraic tensor product. We can identify K with the completion of its everywhere-dense subspace K0. Accordingly, the Hilbert tensor productH⊗F can be viewed as the completion of the algebraic tensor product K0, relative to the unique inner product on K0 that satisfies
< x1⊗y1, x2⊗y2 >=< x1, x2 >< y1, y2 > (x1, x2 ∈H, y1, y2 ∈F). For more details see [9].
3
Frames in Hilbert spaces
Throughout this section V, W, H and F are Hilbert spaces. {Vi : Vi ⊆ V}i∈I
and {Wj :Wj ⊆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 {Λj}j∈J be a g-frame (g-complete, g-Riesz basis) in Hilbert
spaceH with respect to{Vj}j∈J with g-frame operatorS, letF be a Hilbert space
and Q ∈ B(H, F) be invertible. Then {ΛjQ∗}j∈J is a frame(complete,
g-Riesz basis) for F with respect to {Vj}j∈J with g-frame operatorQSQ∗.
More-over if Q is unitary and {Λj}j∈J is a g-orthonormal basis, then {ΛjQ∗}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], {ΛjQ∗}j∈J is a g-frame
with g-frame operator QSQ∗. If {Λj}j∈J is g-complete, then {f : ΛjQ∗f =
0, j ∈J}={f :Q∗f = 0}={0}, becauseQ∗ is one-to-one. Hence {ΛjQ∗}j∈J
is g-complete. If {Λj}j∈J is a g-Riesz basis with bounds A and B, then for
every finite subset J1 of J we have Aj∈J 1 gj 2≤ j∈J1Λ∗jgj 2≤Bj∈J1 gj 2. So A Q−12 j∈J1 gj 2≤ 1 Q−12 j∈J1Λ∗jgj 2 ≤j∈J1QΛ∗jgj 2≤Q2j∈J1Λ∗jgj 2≤B Q2 j∈J1 gj 2.
For the moreover statement, for each j1, j2 ∈ J and xj1 ∈ Vj1, xj2 ∈ Vj2 we
have
< QΛ∗j1xj1, QΛ∗j2xj2 >=< QQ∗Λ∗j1xj1,Λ∗j2xj2 >
=<Λ∗j1xj1,Λ∗j2xj2 >=δj1j2 < xj1, xj2 > and for each
f ∈H ,j∈J ΛjQ∗f 2=Q∗f 2=f 2.
Theorem 3.2 Let H , F, W and V be Hilbert spaces and {Vj}j∈J ⊆ V,
{Wi}i∈I ⊆W be sequences of closed Hilbert subspaces ofV andW, respectively.
Let {Λj}j∈J be a g-orthonormal basis (resp. g-complete, g-Riesz basis) for H
with respect to {Vj}j∈J and {Γi}i∈I be a g-orthonormal basis(resp. g-complete
, g-Riesz basis) for F with respect to {Wi}i∈I. Then {Λj ⊗Γi}i∈I,j∈J is a
g-orthonormal basis (resp. g-complete, g-Riesz basis) for Hilbert space H ⊗F
proof. We know that (Λj ⊗Γi)∗ = (Λ∗j ⊗Γ∗i). Let yj1 ∈ Vj1, yj2 ∈ Vj2 and
xi1 ∈Wi1, xi2 ∈Wi2. Then we have
<Λ∗j1yj1,Λ∗j2yj2 >=δj1,j2 < yj1, yj2 >
and for each y∈H, , j <Λjy,Λjy >=< y, y > . Also
<Γ∗i1xi1,Γ∗i2xi2 >=δi1,i2 < xi1, xi2 >
and for each x∈F, , i <Γix,Γix >=< x, x > .
Therefore <Λ∗j1⊗Γ∗i1(yj1⊗xi1),Λ∗j2⊗Γ∗i2(yj2⊗xi2)> =<Λ∗j1yj1⊗Γ∗i1xi1,Λ∗j2yj2⊗Γ∗i2xi2 > =<Λ∗j1yj1,Λ∗j2yj2 >⊗<Γi1xi1,Γ∗i2xi2 > =δj1,j2δi1,i2 < yj1, yj2>⊗< xi1, xi2 > =δj1,j2δi1,i2 < yj1⊗xi1, yj2⊗xi2 > .
i,j <Λj⊗Γi(x⊗y),Λj ⊗Γi(x⊗y) =i,j <Λjx⊗Γiy,Λjx⊗Γiy >
=i,j <Λjx,Λjx >⊗<Γiy,Γiy >=j <Λjx,Λjx >⊗i <Γiy,Γiy >
=< x, x >⊗< y, y >=< x⊗y, x⊗y > .
Hence relations (iii) hold and therefore{Λj⊗Γi}j∈J,i∈I is a g-orthonormal
basis for H⊗F.
If {Λj}j∈J and {Γi}i∈I are g-complete, then we have {x : Λjx = 0} ={0}
and {y: Γiy= 0} ={0}.SinceΛjx⊗Γiy =ΛjxΓiy= 0 then Λjx= 0
or Γiy = 0 so {x⊗y : (Λj ⊗Γi)(x⊗y) = 0} = {x⊗y : Λjx⊗Γiy = 0} =
{x: Λjx= 0}{y : Γiy = 0}={0}. Hence {Λj⊗Γi}i∈I,j∈J is g-complete.
If {Λj}j∈J and {Γi}i∈I are g-Riesz bases, then there exist constants 0 <
A ≤B <∞such that for each finite subset J1 of J and gj ∈Vj
A j∈J1 < gj, gj >≤ j∈J1 <Λ∗jgj,Λ∗jgj >≤B j∈J1 < gj, gj > .
Also there exist constants 0< C ≤D <∞such that for each finite subset I1 of I and fi ∈Wi we have C i∈I1 < fi, fi >≤ i∈I1 <Γ∗ifi,Γ∗ifi >≤D i∈I1 < fi, fi > .
Therefore for each finite subset J1 of J and I1 of I and every gj ∈Vj and fi ∈Wi we have AC i,j < gj⊗fi, gj⊗fi > = AC i,j < gj, gj >⊗< fi, fi > ≤ C < j Λ∗jgj, j Λ∗jgj >⊗ i < fi, fi > ≤ < j Λ∗jgj, j Λ∗jgj >⊗< i Γ∗ifi, i Γ∗ifi > ≤ B j < gj, gj >⊗D i < fi, fi > = BD i,j < gj⊗fi, gj ⊗fi > .
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, H0 and F0 be Hilbert spaces. Let{Aj}j∈J be an
operator-valued frame onH with range inH0 and{Bi}i∈I be an operator-valued
frame onF with range inF0. Then {Aj⊗Bi}j∈J,i∈I is an operator-valued frame
on H⊗F with range in H0 ⊗F0. In particular, if {Aj}j∈J and {Bi}i∈I are
tight or parseval frames, then so is {Aj⊗Bi}j∈J,i∈I.
Proof. There is a positive bounded invertible operator SA ∈B(H) such
that jA∗jAJ converges in strong operator topology to SA, and there is a
positive bounded invertible operatorSB ∈B(F) such thatiBi∗Bi converges
in strong operator topology to SB. Let x∈H, y ∈F. Then
j∈J1,i∈I1 (Aj⊗Bi)∗(Aj ⊗Bi)(x⊗y) = j∈J1,i∈I1 (A∗jAj ⊗Bi∗Bi)(x⊗y) = j∈J1 A∗jAjx⊗ i∈I1 Bi∗Biy
which converges toSA(x)⊗SB(y) and we have the result. From these it follows
that for all z =kk==1nxk⊗yk in H⊗F,
j∈J1,i∈I1
(Aj ⊗Bi)∗(Aj⊗Bi)(z)−(SA⊗SB)(z)→0
This completes the proof.
Theorem 3.4 Let{Aj}j∈J be an operator-valued frame inB(H, H0)with
anal-ysis operator θA, {Bi}i∈I be an operator-valued frame inB(F, F0)with analysis
operatorθB. ThenθA⊗θB is the analysis operator of the operator-valued frame
proof. As we have in definition 2.4, there are partial isometries{Lj}j∈J from
H0 tol2(J)⊗H0, and {Ki}i∈I fromF0 to l2(I)⊗F0 such that
L∗jLs = I0 (s =j) 0 (s =j) jLjL∗j =I⊗I0 and θA =jLjAj. Similarly Ki∗Ks= I0 (s=j) 0 (s=j)
iKiKi∗ =I ⊗I0 and θB =iKiBi. We define partial isometries
Lj ⊗Ki :H0⊗F0 →l2(J×I)⊗H0⊗F0
, by (Lj ⊗Ki)(h⊗k) = eji ⊗(h⊗k), Where {eji} is the standard basis of
l2(J×I) θA⊗B =j i(Lj⊗Ki)(Aj ⊗Bi) =j iLjAj⊗KiBi =jLjAj ⊗iKiBi =θA⊗θB. We have PA⊗B=θA⊗B(SA⊗B)−1(θA⊗B)∗ = (θA⊗θB)(SA⊗SB)−1(θA⊗θB)∗ =θASA−1θA∗ ⊗θBSB−1θ∗B =PA⊗PB.
Definition 3.5 Let {Aj} and{Bj} be a pair of operator-valued frames on H.
Then {Bj} is called a dual of {Aj} if θ∗BθA = I, the operator-valued frame
{AjSA−1} is called canonical dual frame of {Aj}.
Lemma 3.6 Let {Aj : j ∈ J} and {Bj : j ∈ J} be a pair of dual
operator-valued frames in B(H, H0)and {ak}, {bk} be a pair of dual frames for H0,
re-spectively. Then {A∗jak} and{B∗jbk}are pair of dual frames for H. Moreover,
suppose that {Aj :j ∈J} and {Bj :j ∈J} are canonical dual operator-valued
frames, {ak} , {bk} are canonical dual frames, and that {ak} is a tight frame
with frame bound A. Then {A∗jak} and {Bj∗bk} are canonical dual frames.
proof. {Aj : j ∈ J} is an operator-valued frame, so j∈JA∗jAj = SA
with the frame bounds a and b for which aI ≤SA ≤bI. For every η ∈H, we
have C < Ajη, Ajη >≤ k |< Ajη, ak>|2≤D < Ajη, Ajη > . Consequently C j < Ajη, Ajη >≤ j k |< Ajη, ak >|2≤D j < Ajη, Ajη > .
So C < SAη, η >≤j k|< η, A∗jak >|2≤D < SAη, η >, therefore aC < η, η >≤C < SAη, η >≤j k |< η, A∗jak >|2≤bD < η, η >.
Hence {A∗jak} is a frame for H with frame bounds aC andbD. Similarly
we can get that {Bj∗bk} is a frame for H. On the other hand, for any η ∈H,
we have j k < η, A∗jak > Bj∗bk =jBj∗ k < Ajη, ak> bk = jBj∗Ajη=θ∗θη =Iη =η
Similarly we can get that jk < η, Bj∗bk > Aj∗ak =η. Hence {A∗jak}j,k
and {Bj∗bk}j,k are dual frames forH. Now we assume that {Aj}and {Bj}are
canonical dual operator-valued frames and {ak} is a tight frame with frame
bound A. Then bk = A−1ak. Let SA,a be frame operator associated with
{A∗jak}j,k. Then we have SA,aη= j k < η, A∗jak> A∗jak= j A∗j k < Ajη, ak> ak =A j A∗jAjη=ASAη
Hence SA,a−1A∗jak = A1SA−1A∗jak=Bj∗bk. This completes the proof.
Lemma 3.7 Let{Aj} and {Bj} be a pair of dual operator-valued frames on
H, and let {Ci}, {Di}be a pair of dual operator-valued frames on F. Then
{Aj⊗Ci} and {Bj ⊗Di} are dual operator-valued frames on {H⊗F}.
proof. Since {Aj ⊗Ci} is an operator-valued frame on {H ⊗F} with
θA⊗C =θA⊗θC and {Bj⊗Di}is an operator-valued on{H⊗F}withθ∗B⊗D =
θ∗B⊗θ∗D, then we haveθB∗⊗DθA⊗C = (θ∗B⊗θD∗)(θA⊗θC)=θ∗BθA⊗θD∗θC=IH⊗IF.
This completes the proof.
Definition 3.8 An operator-valued frame {Aj :j ∈J} for which PA =I⊗I0
is called Riesz frame and an operator-valued frame that is both parseval and Riesz, i.e., such that θA∗θA = I and θAθ∗A = I⊗I0, is called an orthonormal
frame.
Lemma 3.9 Let {Aj}j∈J be a Riesz frame(orthonormal frame) in B(H, H0)
and {Bi}i∈I be a Riesz frame(orthonormal frame) in B(F, F0). Then {Aj ⊗
Bi}j∈J,i∈I is a Riesz frame(orthonormal frame) onH⊗F with range inH0⊗F0
proof. Since{Aj}j∈J , {Bi}i∈I are Riesz frames, we have PA=IH ⊗IH0
3.3
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. ρthe right regular representation ofG). Denote by
L(G) ⊆ B(l2(G)), (resp. R(G)) the Von Neumann algebra generated by the unitaries {λg}g∈G, (resp., {ρg}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 χe, where {χg}g∈G is the standard basis of l2(G).
LetH0 be a Hilbert space andI0 be the identity ofB(H0). We callλ⊗id: g ∈G→λg ⊗I0 the left regular representation of Gwith multiplicity H0.
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, H0) is called a frame generator or (resp. a parseval frame generator) with range in H0 for the repre-sentation if {Ag :=Aπg−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 (G1, π1, H) and (G2, π2, F) be unitary representations of discrete groups G1, G2 on Hilbert spaces H, F , with frame generators A and
B, respectively. Then π1 ⊗ π2 : G = G1 ⊕ G2 → B(H ⊗F) is a unitary representation with frame generator A ⊗B. If θA and θB are the analysis
operators for frame generators A and B, respectively, then θA ⊗ θB is the
analysis operator for A⊗B.
proof. G = G1⊕G2, the direct sum of G1 and G2, is a discrete group. We consider the representation π1⊗π2 :G→B(H⊗F), defined by
(π1⊗π2)(g, h) = π1g⊗π2h (g, h)∈G
Since {Ag = Aπ1g−1 : g ∈ G1} is an operator-valued frame for H and {Bh = Bπ2h−1 : h ∈ G2} is an operator valued frame for F, by Theorem 3.3. and the definition of π1 ⊗π2 , {(A⊗B)(π1g−1 ⊗π2h−1) (g, h) ∈ G} is an operator valued frame for H⊗F. Moreover, if θA and θB are the analysis
operators on H and F for frame generators A andB, respectively, then by Theorem 3.4. θA⊗θB is the analysis operator on H ⊗F for frame generator
A⊗B.
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