Hilbert Space Valued Gabor Frames in Weighted Amalgam Spaces 3

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p d 1 d holds for L (R ) and Wiener amalgam spaces (see [16, 24, 25]), and if g,γ ∈ W (Cr,L )(R )

d d then it holds for local Hardy spaces hp(R ), p > d/(d + r) (see [20, 39]), where Cr(R ) denotes the Lipschitz or H¨older space with 0 < r ≤ 1. Moreover, it is desirable to ﬁnd a

Gabor frame such that the generator g and its canonical dual γ have the similar properties

(viz. smoothness or decay) in applications (see [17]). In this direction the following results are known: If g has compact support then in general γ is no longer compactly supported but has exponential decay [8] and in [9] Del Prete proved that if g has exponential decay, then

γ also has exponential decay. If g can be estimated by C(1 + |t|)−s, then the same holds for γ (see [37]). It is natural to ask if g is in a given function space whether its canonical dual is in the same space? The ﬁrst result in this direction is due to Janssen [31] which ensures that if g is in the Schwartz space on Rd then its canonical dual is in the same space.

Gr¨ochenig and Leinert [26] proved that if g is an element in the Feichtinger’s algebra and

G(g,α,β) is a Gabor frame for L2(Rd), then the canonical dual γ is in the same space. In

∞ 1 [33], the authors prove that if G(g,α,β) is a Gabor frame generated by g ∈ W (L ,Lv) then

∞ 1 the Gabor frame operator is invertible on the Wiener amalgam spaces W (L ,Lv), where v is an admissible weight function. In [40], Weisz extended the analogous result on amalgam

∞ q spaces W (L ,Lv) for 1 ≤ q ≤ 2. The above results were based on the reformulation of a non-commutative Wiener’s lemma proved by Baskakov [5, 6]. In [4], Balan et al. obtained a similar result for multi-window Gabor frames on Wiener amalgam spaces using a recent

Wiener type result on non-commutative almost periodic Fourier series [3]. The vector-valued

Gabor frames or superframes were introduced by Balan [1] in the context of “multiplexing”, and several well-known results for Gabor frames are extended to superframes in [2, 29]. The superwavelet and Gabor frames in L2(Rd, Cn) is widely applicable in mathematics and several branches of engineering (see [1, 2, 11, 12, 27, 28, 29, 35, 42, 43]). Using the growth estimates for the Weierstrass σ-function and a new type of interpolation problem for entire functions on

Bargmann-Fock space, Gr¨ochenig and Lyubarskii [27] obtained a complete characterization of all lattices Λ ⊂ R2 such that the Gabor system with ﬁrst n+1 Hermite functions (generators) forms a frame for L2(R, Cn).

The main objective of this article is to investigate the following for superframes on vector- valued amalgam spaces: HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 3

(1) Walnut’s representation for the superframe operator on vector valued amalgam spaces.

(2) Convergence of Gabor expansions on vector valued amalgam spaces.

(3) Invertibility of the superframe operator on vector valued amalgam spaces.

We consider the separable Hilbert space valued Gabor frames and perform Gabor analysis

p q on WH(L ,Lv), 1 ≤ p, q ≤ ∞ to obtain (1), (2) and (3) for Gabor superframes and multi- window Gabor frames as a special case by choosing the appropriate separable Hilbert space

H (see Remark 3.8 and Remark 5.3). We start with the deﬁnition of H−valued Gabor frames on L2(Rd, H).

2 d d Deﬁnition 1.1. Let α,β > 0 and g ∈ L (R , H) be given. For n, k ∈ Z , deﬁne Mβng(x)=

2πihβn,xi e g(x) and Tαkg(x) = g(x − αk). The H-valued Gabor system G(g,α,β) is a frame for L2(Rd, H) if there exist constants A,B > 0 such that for all f ∈ L2(Rd, H),

2 2 2 (1.4) AkfkL2(Rd,H) ≤ |hf,MβnTαkgiL2(Rd,H)| ≤ BkfkL2(Rd,H). Zd k,nX∈ For g, γ ∈ L2(Rd, H), the associated frame operator is given by

2 d Sg,γ f = hf,MβnTαkgiMβnTαkγ, f ∈ L (R , H). Zd k,nX∈ ∞ 1 Moreover if g, γ ∈ WH(L ,Lv), 1 ≤ p, q ≤ ∞, then we obtain Walnut’s representation of the frame operator on H−valued amalgam spaces (see Theorem 3.3):

−d (1.5) S f(x)= β G (x) T n f(x) , g,γ n β Zd nX∈ where Gn(x) is an element of B(H), the class of all bounded linear operators on H. Since we deal with vector-valued functions, obtaining the above expression is bit technical. The expression in Walnut’s representation of the frame operator in (1.5) is similar to scalar-valued case (see [25]), but has a diﬀerent meaning.

1 Further, we show that if the window function g ∈ WH(C0,Lw), the subspace formed by the functions of H−valued Wiener space that are continuous, then the canonical dual is in the same space. To obtain this result we show that the frame operator Sg (= Sg,g) is invertible

p q −1 p q p q on WH(L ,Lv), 1 ≤ p, q ≤ ∞ and Sg : W (L ,Lv) → W (L ,Lv) is continuous both in

′ ′ p q p q σ(W (L ,Lv), W (L ,L1/v)) and the norm topologies. Such invertibility results is addressed for modulation spaces (see [23, 26]) and for Lp spaces (see [33]) by using Wiener’s 1/f lemma

(see [3, 5, 6]). We construct a Banach algebra of operators admitting an expansion like (1.5) 4 ANIRUDHAPORIAANDJITENDRIYASWAIN and use a version of Wiener’s 1/f lemma (see [3]) to prove that the algebra is spectral within the class of bounded linear operators on Lp(Rd, H), 1 ≤ p ≤ ∞.

This paper is organized as follows. In Section 2, we provide basic background and deﬁne

H−valued amalgam spaces. In Section 3, we derive Walnut’s representation for the H-valued

Gabor frame operator and discuss the convergence of H-valued Gabor expansions. In Section

4, we prove the spectral invariance theorem for a subalgebra of weighted-shift operators in

p d B(L (R , H)), and ﬁnally in section 5 we prove the invertibility of the frame operator Sg on

p q WH(L ,Lv), 1 ≤ p, q ≤ ∞.

2. Notations and Background

Throughout this paper, let H denote a separable complex Hilbert space. Let Qα denote

d the cube Qα = [0, α) and χE is the characteristic function of a measurable set E.

Rd p Rd H Deﬁnition 2.1. For 1 ≤ p ≤ ∞ and a strictly positive function w on , let Lw( , ) denote the space of all equivalence classes of H-valued Bochner integrable functions f deﬁned

d p p on R with kf(x)kH w(x) dx < ∞, with the usual adjustment if p = ∞. Rd Z

Deﬁnition 2.2. The Fourier transform of f ∈ L1(Rd, H) is

ˆf(w)= Ff(w)= f(t) e−2πihw,ti dt, w ∈ Rd. Rd Z

′ p d p d Let f ∈ L (R , H). Deﬁne Λf : L (R , H) → C by Λf (g)= hf(x), g(x)iH dx. Then the Rd ′ p d Z p d ∗ map f 7→ Λf deﬁnes an isometric isomorphism of L (R , H) onto [L (R , H)] . For a more detailed study of vector valued functions we refer to Diestel and Uhl [10].

Deﬁnition 2.3. For f, g ∈ L2(Rd, H) the inner product on L2(Rd, H) is deﬁned by

hf, giL2(Rd,H) = hf(x), g(x)iH dx. Rd Z

Deﬁnition 2.4. If x, y ∈ H, the operator x ⊙ y : H → H deﬁned by

(x ⊙ y)(z)= hz,yix, z ∈ H.

Clearly for non-zero x and y, x ⊙ y is a rank one operator with kx ⊙ yk = kxkkyk. HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 5

2.1. Gabor frames in L2(Rd, H). Assume that the Gabor system G(g,α,β) is a frame for

L2(Rd, H) with frame bounds A, B and let h be a vector in H with unit norm. Then the

2 d 2 2d analysis operator is a bounded mapping Cg,h : L (R , H) → ℓ (Z ) deﬁned by Cg,hf =

(hf,MβnTαkgih)k,n∈Zd . The adjoint operator of the analysis operator, called the synthe-

2 2d 2 d sis operator is a bounded mapping Rg,h : ℓ (Z , H) → L (R , H) deﬁned by Rg,hd =

2 hdkn, hiH MβnTαkg. The series deﬁning Rg,hd converges unconditionally in L for ev- k,n∈Zd eryPd ∈ ℓ2. The composition operator of analysis and synthesis operator is called the frame

2 d 2 d operator Sg = Rg,hCg,h : L (R , H) → L (R , H), deﬁned by

Sgf = Rg,hCg,hf = hf,MβnTαkgiMβnTαkg. Zd k,nX∈ The frame operator is strictly positive, invertible, self-adjoint satisfying AIL2(Rd,H) ≤ Sg ≤ −1 −1 −1 −1 2 Rd H BIL2(Rd,H) and B IL2(Rd,H) ≤ Sg ≤ A IL2(Rd,H) . For γ = Sg g ∈ L ( , ), the Gabor expansions

(2.1) Rγ,hCg,hf = hf,MβnTαkγiMβnTαkg, Zd k,nX∈ converge to f unconditionally in L2(Rd, H).

2.2. Weight functions. Weight functions play an important role in time frequency analysis and occur in many problems and contexts. For our study we need the following types of weights.

A function w : Rd → (0, +∞) is called a weight if it is continuous and symmetric (i.e. w(x)= w(−x)). A weight w is submultiplicative if w(x+y) ≤ w(x)w(y), x,y ∈ Rd. Given a submultiplicative weight w, a second weight v : Rd → (0, +∞) is called w-moderate if there exists a constant Cv > 0 such that

d (2.2) v(x + y) ≤ Cvw(x)v(y), x,y ∈ R .

A weight w is called admissible if it is submultiplicative, w(0) = 1 and w satisﬁes the

1/k d Gelfand-Raikov-Shilov condition namely limk→∞ w(kx) =1, x ∈ R .

By (2.2) and by symmetry of w one can conclude that the class of w-moderate weights is

p closed under reciprocals, and consequently the class of spaces Lv using w-moderate weights p Rd H is closed under duality (with the usual exception for p = ∞). Also Lv( , ) is translation- p Rd invariant for moderate weights like Lv( ) (see Proposition 11.2.4. of [30]). Throughout this 6 ANIRUDHAPORIAANDJITENDRIYASWAIN paper, w will denote a submultiplicative weight function and v will denote an w-moderate function. Given an w-moderate weight v on Rd, we will often use the notationv ˜ to denote the weight on Zd deﬁned byv ˜(k) = v(αk), and for a weight v on R2d we deﬁnev ˜(k,n) = v(αk,βn).

2.3. Amalgam spaces.

Deﬁnition 2.5. Given an w-moderate weight v on Rd and given 1 ≤ p, q ≤ ∞, the weighted

p Rd H q H amalgam space W (L ( , ),Lv) is the Banach space of all -valued measurable functions f : Rd → H for which the norm

1/q q q (2.3) kfk p Rd H q := kf · T χ k v(αk) < ∞, W (L ( , ),Lv)  αk Qα Lp(Rd,H)  Zd kX∈   with obvious modiﬁcation for q = ∞.

p q Throughout this paper we denote the weighted amalgam space as WH(L ,Lv) instead of p Rd H q W (L ( , ),Lv). We refer to [13, 14, 15, 19, 30] for discussions of weighted amalgam spaces

p q and their applications. The space WH(L ,Lv) is independent of the value of α used in (2.3)

p q in the sense that each diﬀerent choice of α yields an equivalent norm for WH(L ,Lv) (as in the scalar valued case).

p q Notice that the space WH(L ,Lv) is a Banach function space in the sense of [7]. The K¨othe

p q dual or associated space (see [7]) of WH(L ,Lv) is the space of all measurable functions Rd H 1 Rd p q g : → such that hf(x), g(x)iH ∈ L ( ), for each f ∈ WH(L ,Lv). By Theorem ′ ′ q H p q H p ′ 2.9 of [7], the Köthe dual of W (L ,Lv) coincides with W (L ,L1/v), where 1/p +1/p = ′ 1/q +1/q = 1 for all 1 ≤ p, q ≤ ∞. A series k∈J fk converges (unconditionally) in the ′ ′ q H p q H p P topology of σ(W (L ,Lv), W (L ,L1/v) if the series k∈J hfk, gi converges (is independent ′ ′ q H p P of the ordering of J) for each g ∈ W (L ,L1/v). ′ ′ q H p H p q C H Let f ∈ W (L ,L1/v). Define Θf : W (L ,Lv) → by Θf (g)= hf, gi = Rd hf(x), g(x)i dx. ′ ∗ ′ q H p q R H p Then the map f 7→ Θf defines an isometric isomorphism of W (L ,Lv) onto W (L ,L1/v). We refer to Theorem 11.7.1 of [30] for the duality result in scalar valued amalgam spaces.

We summarize the above discussions in the following lemma. HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 7

Lemma 2.6. Let v be an w-moderate weight and 1/p +1/p′ = 1/q +1/q′ = 1. For 1 ≤

′ ′ q H p q H p H p q p,q < ∞, the dual space of W (L ,Lv) is W (L ,L1/v) and the K¨othe dual of W (L ,Lv) ′ ′ q H p is W (L ,L1/v).

3. H-valued Gabor Expansions in weighted amalgam spaces

The theory of Gabor expansions on Wiener amalgam spaces has been discussed in [16, 21,

24, 25] for a separable lattice Λ = αZd × βZd, α,β > 0. In this section we generalize and extend the Walnut’s representation of the H-valued frame operator and show the convergence

H p q of -valued Gabor expansions on WH(L ,Lv).

1 ˆ Deﬁnition 3.1. The Fourier transform of f ∈ L (Q1/β, H) is the sequence f deﬁned by

ˆf(n)= Ff(n)= βd e−2πiβhn,tif(t) dt, n ∈ Zd. ZQ1/β

p p For 1 ≤ p, q ≤ ∞, denote FL (Q1/β, H) by the image of L (Q1/β, H) under the Fourier

p transform. From the uniqueness of Fourier coeﬃcients for functions in L (Q1/β, H), there p H exists a unique function m ∈ L (Q1/β, ) such that mˆ (n)= dn for every n, if d = (dn)n∈Zd ∈

p p H H p H FL (Q1/β, ). The norm on FL (Q1/β, ) is deﬁned by kdkFL (Q1/β , ) = kmkp,Q1/β .

Deﬁnition 3.2. Let α,β > 0 be given. Let

p,q H Zd p H Sv˜ ( )= {d = (dkn)k,n∈Zd : for each k ∈ , ∃ mk ∈ L (Q1/β, ) such that mˆk(n)= dkn},

1/q q q and kdk p,q H = Zd kmkk v˜(k) < ∞, with the usual change if q = ∞. Sv˜ ( ) k∈ p,Q1/β P 2 H H p 2 Z Since is separable, isometrically isomorphic to ℓ . Therefore kfkL (Q1/β ,ℓ ( )) ≍

p H kfkL (Q1/β , ). When 1

2πiβhn,xi (3.1) mk(x)= dkne , Zd nX∈ p in the sense that the square partial sums of (3.1) converge to mk in the norm of L (Q1/β, H), cf. [36] ([32], [41] for scalar valued case). Hence, for 1 0, 1 ≤ p, q ≤ ∞ and ﬁx g, γ ∈ 8 ANIRUDHAPORIAANDJITENDRIYASWAIN

∞ 1 p q WH(L ,Lw). For f ∈ WH(L ,Lv), deﬁne the analysis operator by

Cg,hf(k,n) = hf,MβnTαkgih

= hf(x),MβnTαkg(x)iHhdx Rd Z −2πiβhn,xi = hf(x),Tαkg(x)iHhe dx Rd Z = F((h ⊙ Tαkg)f)(βn),

2πiβhn,xi where ⊙ is defined in definition 2.4. So n∈Zd hf,MβnTαkgihe = n∈Zd F((h ⊙ 2πiβhn,xi P P 2πiβhn,xi Tαkg)f)(βn)e . Applying Poisson summation formula, n∈Zd hf,MβnTαkgihe = −d n P β n∈Zd ((h ⊙ Tαkg(x))f)(x − β )= mk(x) (say). P p,q H For d = (dkn) ∈ Sv˜ ( ), define the synthesis operator by

Rg,hd(x) = hdkn, hiH MβnTαkg(x) Zd k,nX∈

2πiβhn,xi (3.2) = dkne , h Tαkg(x) Zd * Zd + kX∈ nX∈ H = hmk(x), hiH Tαkg(x) Zd kX∈ 2πiβhn,xi where mk(x)= dkne is 1/β-periodic. n∈Zd From the aboveP observations we obtain the analogue of Walnut’s representation for the

H-valued Gabor frames in the following theorem.

Theorem 3.3. Let α,β > 0 and 1 ≤ p, q ≤ ∞. Let v be an w-moderate weight on Rd and

∞ 1 H g, γ ∈ WH(L ,Lw) with h ∈ be given. If khkH =1 then

p q p,q H (a) The analysis operator Cg,h : WH(L ,Lv) → Sv˜ ( ) deﬁned by Cg,hf = (hf,MβnTαkgih)k,n∈Zd

p is a bounded. There exist unique functions mk ∈ L (Q1/β, H) satisfying mˆ k(n) =

d Cg,hf(k,n) for all k,n ∈ Z which can be expressed explicitly by

−d n −d (3.3) mk(x)= β (h ⊙ Tαkg(x))f(x − )= β (h ⊙ Tαk+ n g(x))T n f(x) β β β Zd Zd nX∈ nX∈ p The series (3.3) converges unconditionally in L (Q1/β, H) (unconditionally in the

∞ 1 topology of σ(L (Q1/β, H),L (Q1/β, H)) if p = ∞). p,q H p q (b) The analysis operator Rg,h : Sv˜ ( ) → WH(L ,Lv) deﬁned by

(3.4) Rg,hd = hmk(·), hiHTakg Zd kX∈ HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 9

p,q H p H is bounded, where d ∈ Sv˜ ( ), mk ∈ L (Qα, ) be the unique functions satisfy-

d ing mˆ k(n) = dkn for all k,n ∈ Z . The series (3.4) converges unconditionally in ′ ′ q H p q H p q H p W (L ,Lv) (unconditionally in the σ(W (L ,Lv), W (L ,L1/v)) topology if p = ∞ or q = ∞).

p q (c) The frame operator Rγ,hCg,h admits Walnut’s representation on WH(L ,Lv):

−d (3.5) R C f = β G T n f γ,h g,h n β Zd nX∈ p q p q holds for f ∈ WH(L ,Lv), with the series (3.5) converging absolutely in WH(L ,Lv), where

(3.6) G (x)= T γ(x) ⊙ T n g(x) ∈ B(H). n αk αk+ β Zd kX∈ We need the following lemma to prove Theorem 3.3.

Lemma 3.4. Let w be a submultiplicative weight, and let α,β > 0 be given. Then there

∞ 1 exists a constant C = C(α, β, w) > 0 such that if g, γ ∈ WH(L ,Lw) and the functions Gn

n are deﬁned by (3.6), then kG k ∞ Rd H w ≤ C kgk ∞ 1 kγk ∞ 1 . n L ( ,B( )) β WH(L ,Lw) WH(L ,Lw) n∈Zd P Proof. Using the fact that w is w-moderate and the translation invariance property for

∞ 1 H ∞ 1 moderate weights (Proposition 11.2.4. of [30]) we get kfwkWH(L ,L ) ≍kfk (L ,Lw). Using the similar idea as in Lemma 6.3.1 of [22] we get the above inequality.

1 d ∞ d Remark 3.5. Let G = {Gn}. Then by the above lemma G ∈ ℓ (Z ,L (R ,B(H))). Then l-th Fourier coeﬃcient of Gn is

−d −2πihl,x/αi Gˆn(l) = α Gn(x)e dx ZQα −d −2πihl,x/αi = α ( T γ(x) ⊙ T n g(x))e dx ak αk+ β Qα Zd Z kX∈ −d −2πihl,x/αi = α (γ(x) ⊙ T n g(x))e dx β Rd Z −d = α γ(x) ⊙ M l T n g(x)dx β Rd α Z −d := α [γ,M l T n g] α β

Then Fourier series

−d 2πihl,x/αi (3.7) Gn(x)= α [γ,M l T n g]e α β Zd lX∈ 10 ANIRUDHAPORIAANDJITENDRIYASWAIN

2 is convergent in L (Qα,B(H)). By substituting this into Walnut’s representation, we obtain the expression

−d −d Sg,γf = β Gn T n f = (αβ) [γ,M l T n g] M l T n f β  α β  α β Zd Zd Zd nX∈ nX∈ lX∈ or in operator notation,  

−d (3.8) Sg,γ = (αβ) [γ,M l T n g] M l T n  α β  α β Zd Zd nX∈ lX∈   This is H-valued analogue of Janssen’s representation for the H-valued frame operator Sg,γ . Using Janssen’s representation we obtain the H-valued analogue of Wexler-Raz biorthogonality relation in the following theorem.

Theorem 3.6. (Wexler-Raz biorthogonality relation) Assume that G(g,α,β), G(γ,α,β) are

Bessel sequence in L2(Rd, H). Then the following conditions are equivalent:

2 d (i) Sg,γ = Sγ,g = I on L (R , H).

−d d (ii) (αβ) [γ,M l T n g]= δl0δn0IB(H) for l,n ∈ Z . α β

Proof. The implication (ii) ⇒ (i) is trivial consequence of Janssen’s representation.

∞ d For the converse (i) ⇒ (ii), assume that Sg,γ = I. Let f, h ∈ L (Q1/β, H) and let l,m ∈ Z be arbitrary. Then

δlm[f, h] = δlm f(x) ⊙ h(x)dx Rd Z = δlm Sg,γf(x) ⊙ h(x)dx Rd Z

= S T l f(x) ⊙ T m h(x)dx g,γ β Rd β Z −d = β G (x) T n+l f(x) ⊙ T m h(x)dx n β Rd β Zd Z nX∈ −d = β G (x) T m f(x) ⊙ T m h(x)dx m−l β β Rd Z −d = β (T m G (x))f(x) ⊙ h(x)dx − β m−l Rd Z −d = β [(T m G )(f), h] − β m−l

2 H −d m By density this identity extends for all f, h ∈ L (Q1/β, ). Thus β Gm−l(x + β ) = d d δlmIB(H) for almost all x ∈ Q1/β. Varying l,m ∈ Z it follows that Gn(x) = β δn0IB(H), for almost all x ∈ Rd. Therefore by (3.7) and uniqueness of Fourier coeﬃcients we have

−d (αβ) [γ,M l T n g]= δl0δn0IB(H) . α β HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 11

Now we are in a position to prove Theorem 3.3.

∞ 1 Proof of Theorem 3.3: (a) Given that g ∈ WH(L ,Lw) and 1 ≤ p, q ≤ ∞. Let f ∈

H p q 1 ∞ W (L ,Lv). Viewing f as a vector in W (L ,L1/w), the series deﬁning mk converges in

1 L (Qα, H). However we show that the series mk converges actually converges unconditionally

p in L (Q1/β, H) (weakly if p = ∞). For each ﬁxed k, we consider

′ |h (h ⊙ T n g(x))T n f(x), h (x)iH|dx αk+ β β Q1/β Zd Z nX∈ ′ = |hf(x),Tαkg(x)iHhh, h (x)iH|Tαk+αnχQα (x)dx Zd Qα nX∈ Z ′ 1/p ′ Cvv(αk + αn)w(αn) ≤ kg · T χ k ∞ Rd H kf · T χ k K kh k ′ αn Qα L ( , ) αk+αn Qα p αβ p ,Q1/β v(αk) Zd nX∈ ′ 1/p ′ 1 = C K kh k ′ kg · T χ k ∞ Rd H w(αn)kf · T χ k v(αk + αn). v αβ p ,Q1/β v(αk) αn Qα L ( , ) αk+αn Qα p Zd nX∈ Taking the supremum in over h′ with unit norm we get,

′ −d 1/p 1 m g ∞ Rd H f k kkp,Q1/β ≤ β CvKαβ v(αk) n∈Zd k · TαnχQα kL ( , )w(αn)k · Tαk+αnχQα kpv(αk + αn), P Zd ℓ where Kαβ = max #{ℓ ∈ : |( β + Q1/β) ∩ (αk + Qα)| > 0}. This shows the convergence of k∈Zd p the series deﬁning mk in L (Q1/β, H). Note that,

d −2πiβhn,xi mˆ k(n) = β mk(x)e dx ZQ1/β −2πiβhn,xi = (h ⊙ T m g(x))T m f(x)e dx αk+ β β Q1/β Zd Z mX∈ = Cg,hf(k,n)

p q p,q H Now we show that Cg,h is a bounded mapping of WH(L ,Lv) into Sv˜ ( ). Deﬁne r(k) = Zd q p,q H kmkkp,Q1/β , k ∈ . We show the sequence r(k) ∈ ℓv˜ which would imply Cg,hf ∈ Sv˜ ( ). ′ q Take the sequence a ∈ ℓ1/v˜. Then we have

|hr, ai| ≤ kmkkp,Q1/β |a(k)| k∈Zd X ′ −d 1/p ∞ ≤ β CvKαβ kg · TαnχQα kL (Rd,H)w(αn) n∈Zd X ′ 1/q 1/q ′ q q q 1 kf · T χ k v(αk + αn) |a(k)| ′  αk+αn Qα p   q  k∈Zd k∈Zd v(αk) X ′ X 1/p −d  ′  (3.9) ≤ β C K kgk ∞ 1 kfk p q kak q . v αβ WH(L ,Lw) WH(L ,Lv) ℓ1/v˜ 12 ANIRUDHAPORIAANDJITENDRIYASWAIN

Taking the supremum over sequences a with unit norm in (3.9) we get kCg,hfk p,q H = Sv˜ ( ) ′ 1/p q −d ∞ 1 q p q krk ≤ β CvK kgk H kfk H p . Hence Cg,h is a bounded mapping of WH(L ,L ) ℓv˜ αβ W (L ,Lw) W (L ,Lv) v p,q H into Sv˜ ( ). This proves (a). (b) Let 1 ≤ p,q < ∞. Given d ∈ Sp,q(H), we have km kq v˜(k)q < ∞. That means v˜ k p,Q1/β k∈Zd for every ε> 0, there exists a ﬁnite set F0 such that P

(3.10) km kq v˜(k)q <εq, ∀ ﬁnite F ⊃ F . k p,Q1/β 0 kX∈ /F ′ ′ q ′ H p Since 1/v is also an w-moderate weight, for any h ∈ W (L ,L1/v), we have

′ |hhmk(·), hiHTαkg, h i| kX∈ /F ′ ≤ |hhmk(x), hiHTαkg(x), h (x)iH|dx Rd kX∈ /F Z ′ = |hmk(x), hiHhTαkg(x), h (x)iH|Tαn+αkχQα (x)dx Zd Qα kX∈ /F nX∈ Z ′ v(αk) ≤ kT g · T χ k ∞ Rd H km k kh · T χ k ′ αk αn+αk Qα L ( , ) k p,αn+αk+Qα αn+αk Qα p v(αn + αk − αn) Zd kX∈ /F nX∈ 1/p ′ Cvv(αk)w(αn) ≤ kg · T χ k ∞ Rd H K km k kh · T χ k ′ . αn Qα L ( , ) αβ k p,Q1/β αn+αk Qα p v(αn + αk) Zd nX∈ kX∈ /F 1/p ′ ′ ′ Using (3.10), we get |hhm (·), hiHT g, h i| ≤ ǫC K kgk ∞ 1 kh k ′ q . k∈ /F k αk v αβ WH(L ,Lw) H p W (L ,L1/v) d Therefore, Rg,hd = P hmk(·), hiHTakg converges unconditionally. Replacing F by Z , we k∈Zd get P

′ ′ |hRg,hd, h i| ≤ |hhmk(·), hiHTαkg, h i| Zd kX∈ 1/p ′ ′ ≤ C K kgk ∞ 1 kdk p,q H kh k ′ q . v αβ WH(L ,Lw) S˜ ( ) H p v W (L ,L1/v)

Thus

1/p q ∞ p,q (3.11) kRg,hdk p ≤ CvK kgk H 1 kdk H , WH(L ,Lv) αβ W (L ,Lw) Sv˜ ( )

′ ′ q H p This completes the proof for the case 1 ≤ p,q < ∞. Since W (L ,L1/v) is the K¨othe dual p q of WH(L ,Lv), a similar argument as in (3.10), (3.11) will imply the convergence of Rg,h in

p q the weak topology when p = ∞ or q = ∞ and the estimate kRg,hdkWH(L ,Lv) as in (3.11). This proves part (b).

p q (c) Now we show the frame operator Rγ,hCg,h admits Walnut’s representation on WH(L ,Lv).

∞ 1 Given g, γ ∈ WH(L ,Lw) and 1 ≤ p, q ≤ ∞. Notice that for a w-moderate weight v and HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 13

p q Zd f ∈ WH(L ,Lv), 1 ≤ p, q ≤ ∞ and for each n ∈ , α> 0 we have

p q p q (3.12) kTαnfkWH(L ,Lv) ≤ Cvw(αn)kfkWH(L ,Lv)

Replacing α by 1/β in (3.12) we get, n kT n fk p q ≤ Cvw( )kfk p q . β WH(L ,Lv) β WH(L ,Lv)

p q Therefore, for f ∈ WH(L ,Lv) consider

kG T n f k p q ≤ kG k ∞ Rd H kT n fk p q n β WH(L ,Lv) n L ( ,B( )) β WH(L ,Lv) n∈Zd n∈Zd X X n ≤ C kfk p q kG k ∞ Rd H w( ) v WH(L ,Lv) n L ( ,B( )) β Zd nX∈ ≤ CC kfk p q kgk ∞ 1 kγk ∞ 1 , v WH(L ,Lv) WH(L ,Lw) WH(L ,Lw)

p q by Lemma 3.4. Therefore the series G T n f converges absolutely in WH(L ,L ). n β v n∈Zd ′ ′ q H p q P ′ H p Now for ﬁxed f ∈ W (L ,Lv), deﬁne mk such that cg,hf(k,n)= mˆ k(n) and h ∈ W (L ,L1/v) we have

′ ′ hRγ,hCg,hf, h i = hhmk(·), hiHTakγ, h i Zd kX∈ ′ = hhmk(x), hiHTakγ(x), h (x)iHdx Rd Zd kX∈ Z −d ′ = β hh(h ⊙ T n g(x))T n f(x), hiHT γ(x), h (x)iHdx αk+ β β ak Rd Zd Zd kX∈ Z nX∈ −d ′ = β hhhT n f(x),T n g(x)iHh, hiHT γ(x), h (x)iHdx β αk+ β ak Rd Zd Zd kX∈ Z nX∈ −d ′ = β hhT n f(x),T n g(x)iHT γ(x), h (x)iHdx β αk+ β ak Rd Zd Zd kX∈ Z nX∈ −d ′ = β h(T γ(x) ⊙ T n g(x))T n f(x), h (x)iHdx ak αk+ β β Rd Zd Zd nX∈ Z kX∈ −d ′ = β hG T n f , h i. n β Zd nX∈ The interchanges of integration and summation can be justiﬁed by Lemma 3.4 and Fubini’s

Theorem. This proves (c).

3.1. Convergence of Gabor expansions.

∞ 1 Proposition 3.7. Let α,β > 0 and 1

(a) the partial sums

SK,N d = hdkn, hiHMβnTαkg, K,N> 0, |kX|≤K |nX|≤N p q p,q H converge to Rg,hd in the norm of WH(L ,Lv), where d ∈ Sv˜ ( ). (b) the partial sums of the Gabor expansion

SK,N (Cg,hf)= hf,MβnTαkgiMβnTαkγ |kX|≤K |nX|≤N ′ ′ q H p q H p q H p converge to f in the norm of W (L ,Lv) and in the σ(W (L ,Lv), W (L ,L1/v))- topology for 1 ≤ p, q ≤ ∞.

Proof. (a) Let ε> 0 be given. Write Rg,hd − SK,N d = (Rg,hd − SK0 d) + (SK0 d − SK0,N d)+

H g (SK0,N d − SK,N d), where SK0 d = |k|≤K0 hmk(·),hi Tαk . Using the boundedness of the p,q H p Pq operator Rg,h : Sv˜ ( ) → WH(L ,Lv) we have 1/q q q q kRg,hd − SK0 dkWH(Lp,L ) = kRg,hk kmkk v˜(k) v  p,Q1/β  |kX|>K0 (3.13) ≤ kRg,hkε. 

Again repeating a similar argument as above and (3.1) we have 1/q q q q kSK0 d − SK0,N dkWH(Lp,L ) ≤ kRg,hk kmk − SN mkk v˜(k) v  p,Q1/β  |kX|≤K0 (3.14) ≤ kRg,hkε, 

2πiβhn,xi p H where SN mk = |n|≤N dkne converging to mk in the norm of L (Q1/β, ), cf. [36]. Finally P 1/q q q q kSK0,N d − SK,N dkWH(Lp,L ) = kRg,hk kSN mkk v˜(k) v  p,Q1/β  K0

d where for each k ∈ Z we can ﬁnd a C1 > 0 satisﬁes

(3.16) sup kSN mkkp,Q1/β ≤ C1kmkkp,Q1/β . N>0

Choose K0,N0 ∈ N such that the inequalities (3.13)-(3.15) are satisﬁed for all K >

p q K0, N>N0 and kRg,hd − SK,N dkWH(L ,Lv) ≤ (2 + C1)kRg,hkε, which completes the proof HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 15 of part (a).

2 d d (b) By (2.1) Rg,hCg,h = I on L (R , H). This would imply Gn(x) = δn0β IB(H) for almost all x ∈ Rd. Using (3.5) and the conclusion of the previous part together proves (b).

The Walnut’s representation for superframe operator and the multi-window Gabor frame operator can be obtained by choosing an appropriate Hilbert space in Theorem 3.3. However, we list out some consequences of Theorem 3.3 and Proposition 3.7 in the following remark.

Remark 3.8. (i) If H = C then the rank one operator x ⊙ y turns out to the point- wise product xy and all the above results for the H-valued Gabor frame viz. Walnut’s representation of H-valued Gabor frame operator, convergence of Gabor expansions, etc., coincides with the results for the scalar valued Gabor frames (see [16, 25, 38, 40]).

(ii) If H = Cn then the H-valued Gabor frame is the super Gabor frame (see [27]). The

Gabor expansions for the Gabor super-frames on vector valued amalgam spaces also converges by Proposition 3.7. H H H H r H (iii) Let ( 1, h·, ·i1), ( 2, h·, ·i2), ··· , ( r, h·, ·ir) be r Hilbert spaces. If = i=1 i (i.e H is the direct sum of r Hilbert spaces) then H is also a Hilbert space with respectL to the r r r H H inner product hx, yi = hxi,yii where x = ⊕i=1xi, y = ⊕i=1yi, x,y ∈ , xi,yi ∈ i,i = i=1 d X 1, 2 ··· , r. If f : R → H then f(x) = f1(x) f2(x) ··· fr(x) with fi(x) ∈ Hi, i =

H p q LH p Lq L 1, 2 ··· , r. Note that f ∈ W (L ,Lw) ⇔ fi ∈ W i (L ,Lw) for all i =1, 2, ··· , r. The frame p q Z Z operator of the Gabor system on WH(L ,Lw) with respect to a single lattice Λ = α × β is given by

−d S f = β G T n f g,γ n β Zd nX∈ −d = β T γ(x) ⊙ T n g(x) T n f αk αk+ β β Zd Zd nX∈ kX∈ r r r −d = β T γ (x) ⊙ T n g (x) T n f αk i αk+ β i β i Zd Zd i=1 ! i=1 ! i=1 !! nX∈ kX∈ M M M r −d = β T γ (x) ⊙ T n g (x) T n f αk i αk+ β i β i Zd Zd i=1 nX∈ kX∈ X r f = Sgi,γi i, i=1 X 16 ANIRUDHAPORIAANDJITENDRIYASWAIN

g r g γ r γ where (x)= i=1 i(x), (x)= i=1 i(x) and Sgi,γi is the frame operator of the Gabor

HL p q H LH H C H system on W i (L ,Lw). If 1 = 2 ··· = r = then the −valued Gabor frame turns out to Gabor superframe.

1 r i d d (iv) Let Λ = Λ × ... × Λ be the Cartesian product of separable lattices Λ = αiZ × βiZ

H ∞ 1 and let g1, ..., gr, γ1, ..., γr ∈ W i (L ,Lw). Suppose the collection Gi(gi, αi,βi) is a frame

i for L2(Rd, H ) with the corresponding frame operator SΛ . As in the pervious set up i gi,γi p q (as in (iii)) we show that frame operator associated with the Gabor system on WH(L ,Lw)

H p q is the sum of frame operator associated with the Gabor systems on W i (L ,Lw). In this

i d d case we consider cartesian product of separable lattices Λ = αiZ × βiZ , i = 1, 2, ··· , r Zd Zd p,q H instead of a single lattice α × β . For 1 ≤ p, q ≤ ∞ the space Sv˜ ( ) turns out to be p,q H p,q H p,q H Sv˜ ( 1) × Sv˜ ( 2) ×···× Sv˜ ( r) with the norm r 1 2 r i kdk p,q H = k(d , d ··· , d )k p,q H = kd k p,q H , Sv˜ ( ) Sv˜ ( ) Sv˜ ( r) i=1 X p,q i where Sv˜ (Hi)is deﬁned as in Deﬁnition 3.2 with respect to the lattice Λ and the Hilbert

d d space Hi. For x ∈ R ,k,n ∈ Z , α = (α1, α2 ··· , αr) and β = (β1,β2 ··· ,βr) with

αi > 0,βi > 0 the translation operator Tαk and the modulation operator Mβn is deﬁned

r r r as Tαkf(x) = i=1 Tαikfi(x) and Mβnf(x) = i=1 Mβinfi(x), where f(x) = i=1 fi(x).

p q The frame operatorL of the Gabor system on WHL(L ,Lw) is given by L

Λ −d S f = β G T n f g,γ n β Zd nX∈ −d = β T γ(x) ⊙ T n g(x) T n f αk αk+ β β Zd Zd nX∈ kX∈ r r r −d n n = β Tαikγi(x) ⊙ Tαik+ gi(x) T fi βi βi Zd Zd i=1 ! i=1 ! i=1 !! nX∈ kX∈ M M M r −d n n = β Tαikγi(x) ⊙ Tαik+ gi(x) T fi βi βi Zd Zd i=1 nX∈ kX∈ X r i = SΛ f , gi,γi i i=1 X i where β−d = r β−d, g(x) = r g (x), γ(x) = r γ (x) and SΛ is the frame i=1 i i=1 i i=1 i gi,γi Q L p q L i operator of the Gabor system on WHi (L ,Lw) with respect to the lattice Λ .

If H1 = H2 ··· = Hr = C and f(x) = f(x) f(x) ··· f(x) (r-times tensor product p q H of f(x) with itself) where f ∈ W (L ,Lw) thenL the L−valuedL Gabor frame turns out to “multi-window Gabor frame”. HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 17

4. The algebra of H-valued L∞-weighted shifts

4.1. H-valued L∞-weighted shifts. In this section we construct a Banach algebra, based on the structure of Walnut’s representation for the frame operator in (3.5) and develop neces-

p q sary tools for invertibility of the frame operator on WH(L ,Lv). For an admissible weight func-

p d tion w we construct a Banach ∗−algebra Aw of weighted shift operators in B(L (R , H)) and

p identify with APw(ρ), the class of ρ-almost periodic elements, having w-summable Fourier

p coeﬃcients. Finally we prove the spectral invariance theorem on APw(ρ) which assures spectral invariance property on Aw. Let us start with a deﬁnition of multiplication operator on

B(Lp(Rd, H)).

∞ d Deﬁnition 4.1. Let φ ∈ L (R ,B(H)), then we deﬁne the multiplication operator Tφ :

p d p d d L (R , H) → L (R , H) deﬁned by (Tφf)(x)= φ(x)(f(x)), x ∈ R .

Note that Tφ is linear, bounded and kTφk = kφkL∞(Rd,B(H)). For an admissible weight w

∞ d deﬁne Aw := {M = (mx)x∈Rd ∈ L (R ,B(H)) : kmxkL∞(Rd,B(H))w(x) < +∞}, with Rd xX∈ ∞ norm kMkAw = kmxkL (Rd,B(H))w(x) < +∞. Rd xX∈ If the family M = (mx)x∈Rd ∈ Aw, then M = (mx)x∈Rd has countable support. The

2 d identiﬁcation of Aw with the subclass of bounded operators on L (R , H) is as follows: Given

p d p d (mx)x∈Rd ∈ Aw, define the operator T : L (R , H) → L (R , H) by T (f) = mx(Txf). x∈Rd Clearly T is well defined, linear and bounded on all Lp(Rd, H), 1 ≤ p ≤P ∞ (by using admissibility of w). The identification f 7→ mx(Txf) maps Aw into a closed x∈Rd p d subspace of B(L (R , H)). We write M ∈ Aw meansP M = mx(Tx), (mx)x∈Rd ∈ x∈Rd 1 Rd ∞ Rd H ℓw( ,L ( ,B( ))). P

2 d If we endow Aw with the product and involution inherited from B(L (R , H)) then Aw is

2 d a Banach *-algebra which embeds continuously into B(L (R , H)): for (mx)x∈Rd , (nx)x∈Rd ∈

Aw, deﬁne x∈Rd mx(Tx) · x∈Rd nx(Tx) = x∈Rd y∈Rd mynx−y(·− y) (Tx), and P P ∗ P P the involution by x∈Rd mx(Tx) = x∈Rd mx(· + x)(T−x) = x∈Rd m−x(·− x)(Tx).

p d Notice that the identiﬁcationP of families P in Aw and operators on BP(L (R , H)) is one-to-

p d one. We simply write Aw ⊂ B(L (R , H)) and we treat members of Aw as operators on

Lp(Rd, H). The following result for m ∈ L∞(Rd,B(H)) plays a crucial role in proving the spectral invariance theorem. 18 ANIRUDHAPORIAANDJITENDRIYASWAIN

Lemma 4.2. For m ∈ L∞(Rd,B(H)) and x, w ∈ Rd. The following relation hold.

2πihw,xi (4.1) Mwm(TxM−w)= e m(Tx)

Proof. The proof of this result is trivial if H = C. Otherwise, for each y ∈ Rd, m(y) is a linear bounded operator on H, which we view as an inﬁnite matrix with scalar entries and prove the lemma. Since H is separable Hilbert space and for y ∈ Rd, m(y) ∈ B(H) can be written as m(y)u = m(y) ( nhu,eniHen)= nhu,eniHm(y)en = n,j hu,eniHanj (y)ej ,

p d where u ∈ H and (en)n is orthonormalP basis for PH. Now for f ∈ L (R P, H)

2πihw,yi Mwm(y)(TxM−wf(y)) = e hTxM−wf(y),eniHanj (y)ej n,j X 2πihw,xi = e hTxf(y),eniHanj (y)ej n,j X 2πihw,xi = e m(y)(Txf(y))

Proposition 4.3. Let 1 ≤ p, q ≤ ∞ and let v be a w-moderate weight. Then

p q (a) the algebra Aw is continuously embedded in B(WH(L ,Lv)). ′ ′ q H p q H p ∗ (b) for every M ∈ Aw, f ∈ W (L ,Lv) and g ∈ W (L ,L1/v), hM(f), gi = hf, M (g)i. ′ ′ q H p q H p Moreover, the operator M is continuous in σ(W (L ,Lv), W (L ,L1/v))-topology.

Proof. The proof of the proposition follows by a straight forward calculation.

p Rd H p 4.2. Spectral invariance. In order to identify the class Aw ⊂ B(L ( , )) with APw(ρ) the class of ρ-almost periodic elements having w-summable Fourier series. We need the following deﬁnitions and necessary theory and apply Theorem 3.2 of [3].

d p d Let 1 ≤ p ≤ ∞,y ∈ R and M ∈ B(L (R , H)). Deﬁne ρ(y)M := MyMM−y. Then,

ρ(y)Mf(x)= e2πihy,xiM(g(x)), where g(x)= e−2πihy,xif(x).

Clearly ρ is a representation of Rd on the Banach space B(Lp(Rd, H)). For each y ∈ Rd, ρ(y) is an algebra automorphism and an isometry.

Deﬁnition 4.4. A continuous map Y : Rd → B(Lp(Rd, H)) is almost-periodic in the sense

d d of Bohr if for every ε> 0 there is a compact set K = Kε ⊂ R such that for all x ∈ R

(x + K) ∩{y ∈ Rd : kY (g + y) − Y (g)k < ε, ∀g ∈ Rd}= 6 ∅ HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 19

ˆd Rd Then Y extends uniquely to a continuous map of the Bohr compactiﬁcation Rc of , ˆd p Rd H ˆd denoted by Y . Thus Y : Rc → B(L ( , )), where Rc represents the topological dual group (i.e. the group of characters) of Rd when Rd is endowed with the discrete topology.

ˆd The normalized Haar measure on Rc is denoted by µ(dy). For each M ∈ B(Lp(Rd, H)), we consider the map M : Rd → B(Lp(Rd, H)) deﬁned by

(4.2) M(y) := ρ(y)M = MyMcM−y.

If the map M is continuous andc almost-periodic in the sense of Bohr then the operator

p Rd H M ∈ B(L ( c, )) is called ρ-almost periodic. For every ρ-almost periodic operator M, the function M admits a B(Lp(Rd, H))-valued Fourier series,

2πihy,xi d (4.3) c M(y) ∼ e Cx(M), (y ∈ R ). Rd xX∈ c p d The coeﬃcients Cx(M) ∈ B(L (R , H)) in (4.3) are uniquely determined by M via

−2πihy,xi 1 −2πihy,xi (4.4) Cx(M)= M(y)e µ(dy) = lim d M(y)e dy ˆd T →∞ (2T ) [−T,T ]d ZRc Z and satisfy c c

2πihy,xi (4.5) ρ(y)Cx(M)= e Cx(M).

Therefore, they are eigenvectors of ρ (see [3] for details).

p Within the class of ρ-almost periodic operators consider APw(ρ), the subclass of those operators for which the Fourier series in (4.3) is w-summable, where w is an admissible

p weight. More precisely, a ρ-almost periodic operator M belongs to APw(ρ) if its Fourier coeﬃcients with respect to ρ satisfy

p p d (4.6) kMkAPw(ρ) := kCx(M)kB(L (R ,H))w(x) < ∞. Rd xX∈ p Since w is submultiplicative, for M ∈ APw(ρ) the series

2πihy,xi d (4.7) M(y)= e Cx(M), y ∈ R , Rd xX∈ c p d p d converges absolutely to M(y) on B(L (R , H)), where each Cx ∈ B(L (R , H)) satisﬁes (4.4)

p and hence (4.5). In particular,c for y = 0, each M ∈ APw(ρ) can be written as

(4.8) M = Cx(M). Rd xX∈ Conversely, if M is given by (4.8) with the coeﬃcients Cx satisfying (4.6) and (4.5), it follows

p from the theory of almost-periodic series that M ∈ APw(ρ) and Cx satisfy (4.4). Now we are 20 ANIRUDHAPORIAANDJITENDRIYASWAIN

p in a position to establish connection between Aw and APw(ρ) and prove spectral invariance result for Aw. For that we ﬁrst characterize the eigenvectors Cx of the representation ρ.

∞ d d Lemma 4.5. For any 1 ≤ p ≤ ∞ and any m ∈ L (R ,B(H)) and x ∈ R , Cx = m(Tx) is an eigenvector of ρ : Rd → B(B(Lp(Rd, H))). For 1 ≤ p< ∞ these are the only eigenvectors.

Proof. If Cx = m(Tx), then by (4.1), it satisﬁes (4.5).

p d For 1 ≤ p < ∞, take Cx ∈ B(L (R , H)) satisfying (4.5). Further the relation (4.1) implies

ρ(y)Cx(T−x)= MyCxM−y(T−x)= Cx(T−x). Therefore Cx(T−xMy)= MyCx(T−x), which in turn imply that Cx(T−x) must be a multiplication operator m. So Cx = m(Tx).

For p = ∞ there are eigenvectors of ρ which may not of the form m(Tx). An example H C p of such an eigenvector is given in ([34], Section 5.1.11) for the case = . Hence APw(ρ) consists of all the operators M = Cx, with Cx satisfying (4.6) and (4.5). The previous x∈Rd lemma says that for 1 ≤ p < ∞ anP operator Cx satisﬁes (4.5) if and only if it is of the

∞ d form Cx = m(Tx), for some function m ∈ L (R ,B(H)). Note that kCxkB(Lp(Rd,H)) = kmkL∞(Rd,B(H)) and thus

∞ d p d p kMkAw = kmxkL (R ,B(H))w(x)= kCx(M)kB(L (R ,H))w(x)= kMkAPw(ρ). Rd Rd xX∈ xX∈ p This gives the identiﬁcation of Aw with APw(ρ).

p Rd H p Proposition 4.6. For p ∈ [1, ∞) the class Aw ⊂ B(L ( , )) coincides with APw(ρ).

For p = ∞, the two classes are diﬀerent. Now we are in a position to prove that the

p d algebra Aw is spectral with in the class of bounded operators on L (R , H). This means if an

p d operator from Aw is invertible on L (R , H) then the inverse operator necessarily belongs to

Aw. In other words invertibility in the bigger algebra implies the invertibility in the smaller algebra. To prove these kind of results one makes use of Wiener’s 1/f lemma or its several versions. We resort to recent Wiener type result on non-commutative almost periodic Fourier series ([3], Theorem 3.2) to obtain the following theorem.

Theorem 4.7. Let w be an admissible weight and M = mx(Tx) ∈ Aw be an invertible x∈Rd p d −1 operator on B(L (R , H)) for some p ∈ [1, ∞] then M ∈P Aw.

Proof. For 1 ≤ p< ∞ the result follows from Proposition 4.6 and Theorem 3.2 in [3]. HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 21

For p = ∞, take

∞ d M = mx(Tx) ∈ Aw ⊂ B(L (R , H)) Rd xX∈ with kmxkL∞(Rd,B(H))w(x) < ∞. Deﬁne x∈Rd P 1 d N = (Tx(m−x))(Tx)= m−x(·− x)(Tx) ∈ Aw ⊂ B(L (R , H)). Rd Rd xX∈ xX∈ Since kTx(m−x)kL∞(Rd,B(H)) = km−xkL∞(Rd,B(H)) N is well deﬁned and a straight forward calculation shows that M is the transpose (Banach adjoint) of N . Therefore N is

1 d −1 invertible when M is invertible. Since Aw is spectral in B(L (R , H)) we get M =

−1 ′ −1 ∞ d (N ) ∈ Aw. That means M = nx(Tx) for some nx ∈ L (R ,B(H)) such that x∈Rd knxkL∞(Rd,B(H))w(x) < ∞. P x∈Rd P 5. H-valued Dual Gabor frames on amalgam spaces

5.1. Invertibility of the frame operators.

∞ 1 Theorem 5.1. Let w be an admissible weight, v be w-moderate weight and g ∈ WH(L ,Lw).

d 2 d Suppose that the Gabor system G(g,α,β)= {MβnTαkg : k,n ∈ Z } is a frame for L (R , H)

−1 p q p q with frame operator Sg. Then inverse operator Sg : WH(L ,Lv) → WH(L ,Lv), 1 ≤ p, q ≤ ′ ′ q H p q H p ∞ is continuous both in σ(W (L ,Lv), W (L ,L1/v)) and the norm topologies.

Proof. As a consequence of the Walnut’s representation in Theorem 3.3 the frame operator

2 d Sg belongs to the algebra Aw. As Sg is invertible in L (R , H), Theorem 4.7 implies that

−1 p q −1 Sg ∈ Aw. Since Aw is continuously embedded in B(WH(L ,Lv)), by Proposition 4.3 Sg ∈ p q B(WH(L ,Lv)). Hence the theorem follows.

d ∞ d Let C0(R ,B(H)) be the subspace formed by the functions of L (R ,B(H)) that are continuous. The next corollary shows the continuity of the dual generator provided the window function is continuous.

Corollary 5.2. If the window function g is continuous then under the assumptions of The-

−1 orem 5.1 the dual window g˜ = Sg (g) is also continuous.

Proof. Let

˜ 1 Rd Rd H Aw = M = (mx)x∈Rd ∈ ℓw( , C0( ,B( ))) | kmxkC0(Rd,B(H))w(x) < ∞ . Rd n xX∈ o 22 ANIRUDHAPORIAANDJITENDRIYASWAIN

p d 1 −d Then A˜ ⊂ A ⊂ B(L (R , H)). If g ∈ WH(C ,L ) then S = β G (T n ) ∈ A˜ . w w 0 w g n β w n∈Zd 2 Rd H ˜ −1 ˜ Since Sg is invertible in B(L ( , )), applying Theorem 4.7 on Aw we getP Sg ∈ Aw.

−1 Let g be continuous. To show Sg (g) is continuous it is enough to show Sg maps

1 1 1 WH(C ,L ) to WH(C ,L ). Let f ∈ WH(C ,L ). Since G (x) T n f(x) is continuous for 0 w 0 w 0 w n β each n, G (x) T n f(x) is continuous. Again, n β ﬁnite P

Gn(x) T n f(x) ≤ kGnk ∞ Rd H kT n f(x)kH β L ( ,B( )) β n∈Zd n∈Zd X H X

∞ ∞ ≤ kGnk Rd H kfkWH(L ,L1 ) < ∞. L ( ,B( )) w Zd nX∈

So by Weierstrass M-test G (x) T n f(x) converges uniformly and hence S f(x) = n β g n∈Zd −d P β G (x) T n f(x) is continuous. n β n∈Zd P p q Remark 5.3. (i) By Proposition 3.7 for all f ∈ WH(L ,Lv)

(5.1) Sg(f) = lim hf,MβnTαkgiMβnTαkg, K,N→∞ kkkX∞≤K knkX∞≤N ′ ′ q H p q H p with convergence in the σ(W (L ,Lv), W (L ,L1/v))-topology and for p,q < ∞ in the norm p q −1 −1 of WH(L ,Lv). Since Sg ∈ Aw, using Proposition 4.3(c) and applying Sg to both sides of (5.1) and we obtain

f = lim hf,MβnTαkgiMβnTαkg˜ K,N→∞ kkkX∞≤K knkX∞≤N = lim hf,MβnTαkg˜iMβnTαkg. K,N→∞ kkkX∞≤K knkX∞≤N p q Similarly using Proposition 3.7 we get the convergence in the norm of WH(L ,Lv). 2 Rd H −1 2 Rd H (ii) If G(g,α,β) is a frame for L ( , ) with dual window g˜ = Sg (g) ∈ L ( , ), then inverse frame operator is given by

−1 Sg,gf = Sg˜,g˜f = hf,TαkMβng˜iTαkMβng˜. Zd Zd kX∈ nX∈ (iii) If H = C then Theorem 5.1 coincides with Theorem 3.2 of [33] and Theorem 2 of [40].

Again if H = Cn the invertibility of Gabor superframes on vector valued amalgam spaces is obtained. If we take H1 = H2 ··· = Hr = C and f(x) = f(x) f(x) ··· f(x) (r-times

p q tensor product of f(x) with itself), where f ∈ W (L ,Lw) asL in RemarkL 3.8L (iv) we obtain the invertibility of multi-window Gabor frames on amalgam spaces (see Theorem 6 of [4]). HILBERT SPACE VALUED GABOR FRAMES IN WEIGHTED AMALGAM SPACES 23

(iv) Since the frame operator Sg ∈ Aw and Sg is invertible, by the last line of the −1 2 Rd H −1 proof of Theorem 4.7, Sg (as an operator on L ( , )), can be expressed as Sg f(x) =

k∈Zd Gn(x)(f(x − xk)) where the family of points {xk} may not lie in the lattice Λ = Pr Zd Zd i=1 αi × βi . Q Acknowledgments

The ﬁrst author wishes to thank the Ministry of Human Resource Development, India for the research fellowship and Indian Institute of Technology Guwahati, India for the support provided during the period of this work. The authors would like to thank the referee for many very helpful comments and suggestions that helped us improve the presentation of this paper.

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Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India. E-mail address: [email protected], [email protected]