Modular Riesz bases versus Riesz bases in Hilbert C ∗ -Modules
aa r X i v : . [ m a t h . F A ] O c t Modular Riesz bases versus Riesz bases inHilbert C ∗ -Modules Marzieh HasannasabOctober 17, 2019
Abstract
In this paper we give new characterizations of modular Riesz basesin Hilbert C ∗ -modules. We prove that modular Riesz bases share manyproperties with Riesz bases in Hilbert spaces. Moreover we show thatthere are also important differences; for example, there exist exactframes that are not modular Riesz bases. The theory of frames in Hilbert C ∗ -modules was introduced by Frank andLarson [6]; more recent works include [3, 5, 7, 8, 11, 14]. Hilbert C ∗ -modulesare generalizations of Hilbert spaces in which the inner product takes values ina C ∗ -algebra. Kasparov’s Stabilization Theorem [9] shows that every finitely orcountably generated Hilbert C ∗ -module has a Parseval frame. The differencesbetween a Hilbert space and a Hilbert C ∗ -module have a significant impacton frame theory in these spaces too. In this paper, we focus on Riesz basesin Hilbert C ∗ -modules. In the literature, two attempts to define Riesz basesin Hilbert C ∗ -modules have been given. In [6], a Riesz basis in a Hilbert C ∗ -module is defined as an ω -independent frame; in [11], modular Riesz bases aredefined by a direct generalization of one of the equivalent definitions of Rieszbases in Hilbert spaces (see the end of the section for the exact definition).We show that modular Riesz bases in Hilbert C ∗ -modules and Riesz bases inHilbert spaces share many key features; indeed, we prove that some of thewell-known characterization theorems for Riesz bases in Hilbert spaces havean analog in Hilbert C ∗ -modules. Furthermore, characterizations of modularRiesz bases with respect to the canonical dual frame and ω -independent framesare given. As the last result we show that even though modular Riesz bases1ehave similar to Riesz bases in Hilbert spaces, there are some importantdifferences due to the special structure of the Hilbert C ∗ -modules. Indeed weshow that Riesz bases and in particular modular Riesz bases in Hilbert C ∗ -modules are exact frames but there exist exact frames in Hilbert C ∗ -modulesthat are not modular Riesz bases.The paper is organized as follows. In the rest of the introduction we willrecall some basic definitions concerning Hilbert C ∗ -modules, their frames andthe two definitions of Riesz bases in such spaces. The new results appear inSection 2.Throughout this paper, A denotes a unital C ∗ -algebra with identity 1 A . Anelement a ∈ A is positive if a = b ∗ b for some b ∈ A . Considering a vector space H , assume that there exists an A -valued inner product h· , ·i : H × H → A with the following properties:(i) h x, x i ≥ x ∈ H and h x, x i = 0 if and only if x = 0,(ii) h x, y i = h y, x i ∗ for every x, y ∈ H ,(iii) h ax + y, z i = a h x, z i + h y, z i for every a ∈ A and x, y, z ∈ H .Under the conditions stated above, H is called a Hilbert A -module (or Hilbert C ∗ -module in case we want to avoid the reference to the name of the underlying C ∗ -algebra) if it is complete with respect to the norm k x k := kh x, x ik . AHilbert A -module H is countably generated if there exists a countable subset { x j : j ∈ N } of H such that the set of all its finite A -linear combinations isdense in H ; in that case the set { x j : j ∈ N } is called a set of generators .Let H and K be two Hilbert A -modules over a C ∗ -algebra A . A linearoperator T : H → K is called adjointable if the exists a linear operator T ∗ : K → H such that h T x, y i = h x, T ∗ y i , for all x ∈ H , y ∈ K . Given any unital C ∗ -algebra, every adjoinable operator is bounded and A -linear, [13]. For every countable index set J, the standard Hilbert A -module isdefined by ℓ ( J, A ) := ( { a j } j ∈ J ⊂ A : X j ∈ J a ∗ j a j is norm convergent in A ) . For each j ∈ J , letting δ ij = 0 if i = j and δ jj = 1, define e j = { δ ij A } i ∈ J .The sequence { e j } j ∈ J is called the canonical orthonormal basis for ℓ ( J, A ).Following [6], we will now give the key definition of frames in Hilbert C ∗ -modules. 2 efinition 1.1 A sequence { x j } j ∈ J of elements in a Hilbert C ∗ -module H issaid to be a frame if the infinite series P j ∈ J h x, x j ih x j , x i converges in normfor all x ∈ H and there exist two constants < C ≤ D < ∞ such that C h x, x i ≤ X j ∈ J h x, x j ih x j , x i ≤ D h x, x i , x ∈ H . (1.1) Appropriate choices for the numbers C and D are called frame bounds. { x j } j ∈ J is a Bessel sequence with bound D if the right-hand side inequality in (1.1) holds. A frame is called tight frame if we can choose C = D and it is calledParseval frame if it is a tight frame with bound 1. Kasparov’s Stabilization Theorem [9] shows that every finitely or countablygenerated Hilbert C ∗ -module over a unital C ∗ -algebra has a frame. In in-equality (1.1) we are comparing the positive elements in A . Arambaˇsi´c [1] andJing in [8], independently showed that one can replace (1.1) with two inequal-ities in terms of the norm of elements. Indeed, it is proved that a sequence { x j } j ∈ J ⊂ H is a frame if and only if there exist positive constants C and D such that C k x k ≤ k X j ∈ J h x, x j ih x j , x ik ≤ D k x k , x ∈ H . For every Bessel sequence { x j } j ∈ J , the operator T : H → l ( J, A ) defined as T x = {h x, x j i} j ∈ J , x ∈ H is called the analysis operator . The operator T is adjointable and its adjoint,the synthesis operator , is given by U { a j } j ∈ J = P j ∈ J a j x j . The frame operator S : H → H , defined as Sx = U ∗ U ( x ) = P j ∈ J h x, x j i x j , is bounded, positiveand invertible, see [6]. Thus the following reconstruction formula holds forframes in Hilbert C ∗ -modules: x = SS − x = X j ∈ J h S − x, x j i x j = X j ∈ J h x, S − x j i x j , x ∈ H . (1.2)We call { S − x j } j ∈ J the canonical dual frame of { x j } j ∈ J . Also if { x j } j ∈ J and { y j } j ∈ J are frames and x = P j ∈ J h x, y j i x j , for all x ∈ H , then { x j } j ∈ J and { y j } j ∈ J are called dual frames .Following [6], a frame { x j } j ∈ J for a Hilbert A -module H is a Riesz basis ifthe following two conditions are satisfied:(i) x j = 0 for all j ∈ J . 3ii) If P j ∈ S a j x j = 0 for a finite set of coefficients { a j } j ∈ S ⊂ A , S ⊆ J , then a j x j = 0 for every j ∈ S .It is proved in [7] that a Riesz basis in a Hilbert C ∗ -module may have manydual frames and it may even admit two different dual frames both of whichare Riesz bases. This shows that the definition of Riesz bases in Hilbert C ∗ -modules is not the analog of Riesz basis in Hilbert spaces. Later, in [10]modular Riesz bases were introduced: Definition 1.2
A sequence { x j } j ∈ J in Hilbert A -module H is called a modularRiesz basis for H , if there exists an invertible A -linear and adjointable operator U : l ( J, A ) → H such that U e j = x j for each j ∈ J , where { e j } j ∈ J = { ( δ ij A ) i ∈ J } j ∈ J is the standard orthonormal basis of l ( J, A ) . It is proved in [11] that a sequence { x i } j ∈ J ⊂ H is a modular Riesz basis ifand only if { x j } j ∈ J is a set of generators for H (as a Banach A -module) andthere exist C, D > { a i } i ∈ S in A , C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ S | a i | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ S a i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ D (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ S | a i | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (1.3)This gives a characterization of modular Riesz bases in terms of the analysisoperator. As a consequence of (1.3), it is proved in [11] that the set of allmodular Riesz bases coincides with the set of all frames that have a uniquedual frame which is a modular Riesz basis. Also it is straightforward from(1.3), that the analysis operator of a modular Riesz basis is bijective and thatan infinite series P j ∈ J a j x j is convergent if and only if { a j } j ∈ J ∈ ℓ ( J, A ). It is well-known that in a Hilbert space H , Riesz bases, w -independent frames,and exact frames are different names for the same class of sequences. Wewill now define the analogue version of the mentioned sequences in Hilbert C ∗ -modules, and consider their interrelations. Given a C ∗ -algebra A and aHilbert A -module H , let A -span { x j } j ∈ J := (X j ∈ S a j x j : a j ∈ A , and S ⊂ J is a finite subset ) . efinition 2.1 A sequence { x j } j ∈ J ⊂ H is said to be:(i) ω -independent, if whenever P j ∈ J a j x j is convergent and equal to zero forsome { a j } j ∈ J ⊂ A , then necessarily a j = 0 for all j ∈ J .(ii) biorthogonal with { y j } j ∈ J ⊂ H , if h x i , y j i = δ ij A for all i, j ∈ J. (iii) exact frame, if { x j } j ∈ J is a frame and for every ℓ ∈ J , { x j } j = ℓ cease tobe a frame for H . The following result generalizes Theorem 7.7.1 in [4] to Hilbert C ∗ -modulesand shows the connection between the sequences defined in part ( i ) and ( ii )of Definition 2.1 and modular Riesz bases. Theorem 2.2
Let { x j } j ∈ J be a frame in a Hilbert A -module H . Then thefollowing statements are equivalent:(i) { x j } j ∈ J is a modular Riesz basis,(ii) { x j } j ∈ J is ω -independent,(iii) { x j } j ∈ J and its canonical dual { S − x j } j ∈ J are biorthogonal,(iv) { x j } j ∈ J has a biorthogonal sequence, Proof. ( i ) ⇒ ( ii ) If { x j } j ∈ J is a modular Riesz basis, then T ∗ : l ( J, A ) → H is bijective by (1.3). Assume that P j ∈ J a j x j = 0 for some { a j } j ∈ J ⊆ A . Since P j ∈ J a j x j is convergent, { a j } j ∈ J ∈ l ( J, A ). Therefore { a j } j ∈ J ∈ KerT ∗ .Since T ∗ is injective, we have { a j } j ∈ J = 0.( ii ) ⇒ ( iii ) Consider the canonical dual frame { S − x j } j ∈ J , where S is theframe operator of { x j } j ∈ J . We have x ℓ = P j ∈ J h S − x ℓ , x j i x j for all ℓ ∈ J .Hence X j = ℓ h S − x ℓ , x j i x j + ( h S − x ℓ , x ℓ i − A ) x ℓ = 0 . Since { x j } j ∈ J is ω -independent, we have h S − x ℓ , x ℓ i = 1 A , h S − x ℓ , x j i = 0 f or each j = ℓ. ( iii ) ⇒ ( iv ) Clear.( iv ) ⇒ ( i ) Let T ∗ be the synthesis operator for { x j } j ∈ J and let { a j } j ∈ J ∈ KerT ∗ and { y j } j ∈ J be biorthogonal to { x j } j ∈ J . Then for every ℓ ∈ J , a ℓ = h a ℓ x ℓ , y ℓ i = h X j ∈ J a j x j , y ℓ i = 0 . Therefore the synthesis operator is bijective. (cid:3) orollary 2.3 Let H be a Hilbert A -module H over a unital C ∗ -algebra A and let { x j } j ∈ J be a frame in H with the canonical dual frame { S − x j } j ∈ J . If h x j , S − x j i = 1 A for each j ∈ J , then { x j } j ∈ J is a modular Riesz basis. Proof.
By the reconstruction formula, for each j ∈ J we have x j = X ℓ ∈ J h x j , S − x ℓ i x ℓ = x j + X ℓ = j h x j , S − x ℓ i x ℓ . Thus P ℓ = j h x j , S − x ℓ i x ℓ = 0. Applying S − , we obtain P ℓ = j h x j , S − x ℓ i S − x ℓ =0. From this relation, we conclude that X ℓ = j h x j , S − x ℓ ih S − x ℓ , x j i = 0 , for all j ∈ J. (2.1)Since the element h x j , S − x ℓ ih S − x ℓ , x j i = h x j , S − x ℓ ih x j , S − x ℓ i ∗ is positivefor all j, ℓ ∈ J , (2.1) implies that kh x j , S − x ℓ ik = kh x j , S − x ℓ ih x j , S − x ℓ i ∗ k = 0 for all ℓ ∈ J \ { j } . This means that { x j } j ∈ J and its canonical dual frame are biorthogonal. There-fore by Theorem 2.2, { x j } j ∈ J is a modular Riesz basis. (cid:3) In a Hilbert space, a frame is exact if and only if it is a Riesz basis, [4,Theorem 5.5.4]. We will now analyse the relationship between the exact framesand Riesz bases in Hilbert C ∗ -modules; in particular the result with show thatthe situation is different compare to the Hilbert space case. We will need thefollowing lemma, which yields a characterization for exact frames in Hilbert C ∗ -modules, see [8] for the proof. Lemma 2.4
Let { x j } j ∈ J be a frame for a Hilbert A -module H and let A bethe identity element of A . For each ℓ ∈ J , the sequence { x j } j = ℓ is a frame for H if and only if A − h x ℓ , S − x ℓ i is invertible in A . The following result shows that not only modular Riesz basis but also theRiesz bases are indeed exact frames.
Proposition 2.5
Let H be a Hilbert A -module H over a unital C ∗ -algebra A .If { x j } j ∈ J is a Riesz basis then it is an exact frame. Proof.
Assume that { x j } j ∈ J is a Riesz basis and that there exists some ℓ ∈ J such that { x j } j = ℓ is a frame for H . Then there exists { a j } j ∈ J \{ ℓ } ⊂ A suchthat x ℓ = P j = ℓ a j x j . Hence letting a ℓ = − A , we have P j ∈ J a j x j = 0. Thisleads to a contradiction with the definition of a Riesz basis. (cid:3) Example 2.6
Consider the Banach space of all bounded sequences ℓ ∞ ( N ).This space forms a C ∗ -algebra with respect to operations { u j } j ∈ N . { v j } j ∈ N = { u j v j } j ∈ N , { u j } ∗ j ∈ N = { ¯ u j } j ∈ N . Let H = c denote the Banach space of all sequences vanishing at infinity.Then H is a Hilbert A -module with inner product h { u j } j ∈ N , { v j } j ∈ N i = { u j ¯ v j } j ∈ N , { u j } j ∈ N , { v j } j ∈ N ∈ H . Let δ j ∈ H be the sequence in H that takes the value 1 at the jth coordinateand 0 everywhere else. Note that the sequence { δ j } j ∈ J is a Parseval frame,since for every u = { u j } j ∈ J ∈ H (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ∈ J h u, δ j ih δ j , u i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ∈ J | u j | δ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13) {| u j | } j ∈ J (cid:13)(cid:13) = sup | u j | = (sup | u j | ) = k u k . Thus the frame operator is S = Id H and for each j ∈ N , h δ j , S − δ j i = h δ j , δ j i = δ j . Therefore the sequence 1 A − h δ j , S − δ j i takes the value 0 at the jth coordinateand 1 everywhere else. This shows that h δ j , S − δ j i 6 = 1 A and that also 1 A −h δ j , S − δ j i is not invertible in A . By Lemma 2.4 we conclude that { δ j } j ∈ N isan exact frame. On the other hand Theorem 2.2 implies that { δ j } j ∈ N is not amodular Riesz basis. Moreover, the set { δ j } j ∈ N is indeed a Riesz basis for H .To see this, let a j = { a ij } i ∈ N ∈ A and S ⊂ N be a finite set and assume that P j ∈ S a j δ j = 0. We have X j ∈ S a j δ j = X j ∈ S { a ij δ ij } i ∈ N = { X j ∈ S a ij δ ij } i ∈ N = 0 . This implies that a jj = 0 for all j ∈ S and therefore a j δ j = 0. This shows that { δ j } j ∈ N is a Riesz basis for H . (cid:3) cknowledgments After acceptance of the paper the author became award of the paper [2] whichhas some overlap with the results. Indeed, part of the results in Theorem 2.2also follows from Cor. 4.20 in [2]. The author would like to thank ProfessorArambasic for pointing this out.
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