Unitary groups and spectral sets

# Unitary groups and spectral sets

Dorin Ervin Dutkay [Dorin Ervin Dutkay] University of Central Florida
Department of Mathematics
4000 Central Florida Blvd.
P.O. Box 161364
Orlando, FL 32816-1364
U.S.A.
and  Palle E.T. Jorgensen [Palle E.T. Jorgensen]University of Iowa
Department of Mathematics
14 MacLean Hall
Iowa City, IA 52242-1419
###### Abstract.

We study spectral theory for bounded Borel subsets of and in particular finite unions of intervals. For Hilbert space, we take of the union of the intervals. This yields a boundary value problem arising from the minimal operator with domain consisting of functions vanishing at the endpoints. We offer a detailed interplay between geometric configurations of unions of intervals and a spectral theory for the corresponding self-adjoint extensions of and for the associated unitary groups of local translations. While motivated by scattering theory and quantum graphs, our present focus is on the Fuglede-spectral pair problem. Stated more generally, this problem asks for a determination of those bounded Borel sets in such that has an orthogonal basis of Fourier frequencies (spectrum), i.e., a total set of orthogonal complex exponentials restricted to . In the general case, we characterize Borel sets having this spectral property in terms of a unitary representation of acting by local translations. The case of is of special interest, hence the interval-configurations. We give a characterization of those geometric interval-configurations which allow Fourier spectra directly in terms of the self-adjoint extensions of the minimal operator . This allows for a direct and explicit interplay between geometry and spectra.

###### Key words and phrases:
Fuglede conjecture, unbounded operators, domains, adjoints, spectrum, deficiency-indices, Hermitian operators, self-adjoint extensions, unitary one-parameter groups, spectral pairs, boundary values, reproducing kernel Hilbert spaces, scattering theory, locally compact Abelian groups, Fourier analysis.
###### 2000 Mathematics Subject Classification:
Primary 47B25 , 47B15, 47B32, 47B40, 43A70, 34K08. Secondary 35P25, 58J50.
\@footnotetext

## 1. Introduction

In this paper we study a classification problem (SAE) for self-adjoint extensions of Hermitian operators in Hilbert space, with dense domain and finite deficiency indices . While this question has many ramifications, we will focus here on a restricted family of extension operators. We begin with a justification for the restricted focus.

Definitions. It turns out that this problem arises in a number of instances which on the face of it appear quite different, but turn out to be unitarily equivalent. While we have in mind models for scattering of waves on a disconnected obstacle, and quantum mechanical transition probabilities, it will be convenient for us to select the version of problem (SAE) where and is a bounded open subset of the real line with a finite number of components, i.e., is a finite union of open disjoint intervals. We consider corresponding to vanishing boundary conditions (the minimal operator). Then the deficiency indices are when is the number of components in . In a different context, mathematical physics, the minimal operator was considered in [Jør81].

Let be a complex Hilbert space with inner product . Let be a dense subspace in . A linear operator defined on is said to be symmetric (or Hermitian) iff

 ⟨Lf,g⟩=⟨f,Lg⟩ % for all f,g∈D.

In this case, the adjoint operator is defined on a subspace containing and , where ”” refers to containment of graphs.

If the dimensions of the two eigenspaces are equal (called the deficiency indices ) then has self-adjoint extensions. Every self-adjoint extension of must satisfy and any such will be a restriction of .

In section 3, we offer a geometric model for the study of finite deficiency indices . While this work is directly related to recent work [JPT12c, JPT12d, JPT12a], our present focus is different, as are our themes. To make our present paper reasonably self-contained, it will be convenient for us to include here (section 3) some basic lemmas needed in the proof of our main theorems. When (the number of intervals in ) is fixed, the set of all self-adjoint extensions of is in bijective correspondence with the group of all complex unitary matrices. Moreover we include in Proposition 3.14 an explicit formula for our correspondence for the problem (SAE), expressed directly in terms of the interval-endpoints constituting the boundary of . It is an action by elements in the conformal group .

Motivation. One motivation for our study is a spectral theoretic question (conjecture) raised first in a paper by Fuglede [Fug74]. We refer to [Fug74] and [JP96] for details, but, in summary, the question is whether the existence of a Fourier basis in is equivalent to the set possibly tiling by a set of translation vectors. Here the question addresses any dimension , and for any Borel set of finite positive Lebesgue measure. If there is a subset in such that the complex exponentials form an orthogonal basis in we say that is a spectral pair, and we say that the set is spectral.

There are a number of reasons for restricting to a one-dimensional model.

In application to the Lax-Phillips model for scattering of acoustic waves on a bounded and disconnected obstacle in , (see e.g., [PWW87]), the present case of one dimension will then represent a wave motion in a single direction in . In the Lax-Phillips model for obstacle scattering, time-evolution of waves is represented by a unitary one-parameter group of operators acting on an associated energy Hilbert space .

Below we outline briefly two reasons why our results about intervals are relevant to Lax-Phillips scattering theory [LP89]. Recall that Lax-Phillips scattering theory deals with scattering of acoustic waves around solid obstacles in , i.e., the solution to some wave equation, represented in a Hilbert space, and the study of wave solutions in the complement of a compact obstacle, e.g., for the wave equation in an exterior domain. This study of the exterior of a bounded obstacle in turn is divided into the case of a connected obstacle, vs the case of disconnected ones. The latter is more subtle because it yields intriguing configurations of bound-states, i.e., the trapping of waves in the bounded connected components in the complement obstacle: in mathematical language, eigenvectors and eigenvalues. Now working directly in and in components in the complement of a bounded obstacle is typically difficult, both in the theory and in applications. One way around this difficulty goes via hyperplane segments in the obstacle, e.g., the use of a suitable Radon transform [Hel11]. But rather, our present approach is to study instead waves traveling along varying linear directions in . These directions in turn are specified by lines passing through the obstacles.

With disconnected obstacles in , the linear intersections will then be the union of a number of bounded intervals. Using a second fact from Lax-Phillips scattering theory [LP89], we note that the solutions to the acoustic wave equations may be represented by a strongly continuous unitary one-parameter group, say acting on an energy Hilbert space. But Lax and Phillips [LP89] show that this group may be taken to be unitarily equivalent to a translation representation. Hence, for each of the one-dimensional linear directions, we get an equivalent translation group generated by a self-adjoint operator arising in a one-dimensional boundary value problem; and hence we are led to self-adjoint extensions of a minimal derivative in some linear set that is the union of a finite number of disjoint open intervals. Note that will vary with the choice of different linear directions through some fixed disconnected obstacle.

Starting with one fixed such set , we study the question of self-adjoint extensions. While our motivation derives from the problem of spectral pairs, the problem is of independent interest in scattering theory. And even in the study of possible spectral sets , one must consider the variety of all self-adjoint extensions, although only a few of these extensions, if any, yield spectral pairs. If some is spectral, we show that as the spectrum part in a corresponding spectral pair, one may take for the spectrum of some self-adjoint extension. Only with hindsight one realizes that in fact “most” self-adjoint extensions are not spectral. Identifying those that are is a subtle problem.

Our main results are in sections 2 through 4, and we give an application in section 5. Some highpoints: In Theorem 2.2 we show that every spectral pair , with a bounded Borel set of positive Lebesgue measure, has a group of local translations. In Theorem 2.12 we apply this to the case when has a spectrum with period , where is a fixed positive integer. In Sections 3 and 4 we turn to the cases when is assumed a union of a finite number of non-overlapping intervals. We introduce determinant-bundles, and we use them in Theorem 3.15 in order to classify the spectral types of self-adjoint extensions. This, and our local translation groups, are applied, in turn, in proving our results on spectral correspondence: In Theorem 4.9 and Corollary 4.10 we offer a classification for the case when the spectrum has period ; and in Theorems 4.13 and 4.14 we further study the correspondence between two sets making up a spectral pair.

We view the approach to spectral pairs via self-adjoint extensions as a tool for generating spectra. In fact, in the literature, so far there are rather few analytic and constructive tools available helping one produce spectral pairs. We present an analysis of self-adjoint extension as one such tool. Our purpose is to develop these self-adjoint extensions, refine them, and apply this to the spectral pair question. But these other applications to scattering theory are of general interest in mathematical physics.

As for spectral pairs, we find that among all the self-adjoint extensions only a very small subset is spectral. And of course, for many cases of a linear set , this (spectral) subset may be empty. But even if some is not spectral, we have a detailed geometric configuration of self-adjoint extensions with associated spectra. We also study these, their nature and geometry.

Now, Fuglede’s conjecture is known to be negative when the dimension is 3 or more [Tao04, KM06] , but the cases of and are still open. But in the plane () the partial derivatives corresponding to zero boundary conditions for a fixed open planar are known to have deficiency indices . Moreover the geometric issues involved for the two cases and are quite different, and we thus focus here on .

We prove in section 3 the following theorem: Given a finite number of components in some fixed , and a pair, consisting of a matrix and boundary points; i.e., given in , and interval endpoints (the boundary of ), by passing to the corresponding self-adjoint extension in we get a spectral pair (, spectrum()) if and only the matrix has a certain factorization in terms of two unitary matrices, one formed from the left-hand side interval endpoints, and the other from the right-hand side interval endpoints.

There is a recent substantial prior literature on orthogonal Fourier exponentials and spectral duality, see for example [Fug74, DJ09, DHL09, DJ08, DJ07a, DJ07b, JP98, JP96, JP99, LS92, IP98, IKT01, Tao04, KM06, BJ11, JPT12a, JPT12b].

###### Definition 1.1.

For , we denote by , . A Borel subset of finite Lebesgue measure is said to be spectral if there is a set in such that the family of exponential functions is an orthonormal basis for . In this case, is called a spectrum for and is called a spectral pair.

indicates the Lebesgue measure of .

A finite Borel measure on is called spectral if there exists a set in such that is an orthogonal basis for . We call a spectrum for .

A finite set in is spectral if the atomic measure is spectral.

A Borel subset of tiles by translations if there exists a subset of such that forms a partition of , up to Lebesgue measure zero.

Main results and the structure of the paper. In section 2, we turn to the study of general spectral subsets of the line, i.e., for . In Theorem 2.2, we prove that if some set is a first part in a spectral pair then there is a canonically associated unitary one-parameter group consisting of local translations in . If is further assumed open, then this becomes a statement about the infinitesimal generator of as a self-adjoint extension of the minimal operator for .

Section 2 contains a number of additional detailed results. We highlight the following: in Theorem 2.12, for the general case of linear spectral sets , we offer a geometric representation of the associated unitary one-parameter group of local translations in : We show that this one-parameter group is unitarily equivalent to an induced representation of in the sense of Mackey [Mac62].

In section 3, we turn to a detailed analysis of the set of all self-adjoint extensions of the minimal operator for a fixed bounded open linear set written as a union of a finite number of components, see Definition 3.1. We show in Theorem 3.7 that self-adjoint extensions of the minimal operator correspond to unitary matrices . This follows the same pattern as the ones in [JPT12c, JPT12d, JPT12a], but here the intervals are all finite. In addition, in Proposition 3.12 we describe the reproducing kernel Hilbert space structure for the graph-inner product and offer formulas for the kernel functions. In Theorem 3.15 we describe the spectral decomposition of the self-adjoint extensions in terms of the unitary matrix .

Section 4 deals with spectral sets which are finite unions of intervals. In Theorem 4.1 we show that some finite union of intervals in is spectral if and only if there is a strongly continuous unitary one-parameter group acting in the Hilbert space by pointwise translation inside , i.e., sending points in to whenever both are in . This extends, in dimension one, results from [Fug74, Ped87] (the results due to Fuglede and Pedersen are formulated in terms of the “integrability property” for , which is similar to our local translation property but the translations are made only with small enough numbers; also the equivalence between the spectral property and the integrability property is true only for connected sets).

In Theorem 4.4 and Corollary 4.5, for a given , assumed spectral, we characterize those self-adjoint extensions of the minimal operator which correspond to spectral pairs and moreover, we give a formula for the corresponding spectrum . Our Theorems 4.9, 4.13, and 4.14 together offer a geometric properties of spectral sets .

Finally in section 5, as an illustrating example, we specialize to the case when is the disjoint union of two disjoint open intervals. While this may appear overly specialized, we stress that it is of significance in the above mentioned applications to Lax-Phillips scattering theory. Part of these results can be found in [Łab01, JPT12d], but we include here more detailed description including one for the associated groups of local translations.

## 2. General spectral subsets Ω of R

In this section we study the case of spectral pairs for bounded Borel subsets of ; a key ingredient is a result from [BM11, IK12] that the spectrum has a finite period. But first we show that the spectral property implies the existence of a certain unitary group of local translations and we give a detailed description of this unitary group and various equivalent forms.

###### Definition 2.1.

Let be a bounded Borel subset of . A unitary group of local translations on is a strongly continuous one parameter unitary group on with the property that for any and any ,

 (2.1) (U(t)f)(x)=f(x+t) for a.e x∈Ω∩(Ω−t)

If is spectral with spectrum , we define the Fourier transform

 (2.2) Ff=(⟨f,1√|Ω|eλ⟩)λ∈Λ,(f∈L2(Ω)).

We define the unitary group of local translations associated to by

 (2.3) UΛ(t)=F−1^UΛ(t)F where ^UΛ(t)(aλ)=(e2πiλtaλ),((aλ)∈l2(Λ).
###### Theorem 2.2.

Let be a bounded Borel subset of . Assume that is spectral with spectrum . Let be the associated unitary group as in (2.3). Then is a unitary group of local translations.

###### Proof.

We will show that (2.1) holds. First note that for and . So for and we have

 (U(t)eλ)(x)=e2πiλte2πiλx=e2πλ(x+t)=eλ(x+t),

hence (2.1) holds everywhere for .

Let . Since the set is an orthogonal basis for , we can approximate in by a sequence of functions which are finite linear combinations of the functions , . By passing to a subsequence we can also assume that converges to for where has Lebesgue measure zero. Since is unitary, we have that converges to , and again by passing to a subsequence we can assume that converges to for where has Lebesgue measure zero.

The set has Lebesgue measure zero. Take . We have and . Also

 (U(t)fn)(x)=fn(x+t)→f(x+t)

At the same time converges to . Thus we have for .

Our next goal is to give a more precise description of the group of local translations associated to a spectrum. One of the main ingredients that we will use is the fact that any spectrum is periodic (see [BM11, IK12]).

###### Definition 2.3.

If , is spectral then any spectrum is periodic with some period , i.e., , and is an integer multiple of . We call a period for . If with , then has the form

 (2.4) Λ={λ0,…,λk(p)−1}+pZ,

with , see [BM11, IK12]. The reason that there are elements of in the interval can be seen also from the fact that the Beurling density of a spectrum has to be , see [Lan67].

###### Remark 2.4.

According to [IK12], if has Lebesgue measure 1 and is spectral with spectrum , with , then is periodic, the period is an integer and has the form

 (2.5) Λ={λ0=0,λ1,…,λp−1}+pZ,

where in are some distinct real numbers.

We recall a few lemmas and propositions (see [DJ12a] and the references therein) that exploit the periodicity of the spectrum to give some information about the structure of .

###### Proposition 2.5.

If , is spectral, and for all , then any spectrum for has as a period.

###### Corollary 2.6.

If , is spectral and for all and if a spectrum has period then divides .

###### Proposition 2.7.

Let be a bounded Borel set of measure 1. Assume that is spectral with spectrum , , which has period . Then is a -tile of by -translations, i.e., for almost every , there exist exactly integers such that is in , .

Also, for a.e. in , there are exactly integers such that is in for all .

###### Definition 2.8.

For , define the multiplication operator on by , .

###### Lemma 2.9.

Let be a bounded Borel set of measure 1. Assume that is spectral with spectrum , which has period . Let be the orthogonal projection in onto the closed subspace spanned by . Then, for ,

 (2.6) (P(pZ)f)(x)=1p∑j∈Zf(x+jp), for a.e. x∈Ω.

(We define to be zero outside )

###### Proposition 2.10.

Let be a bounded Borel set of measure 1. Assume that is spectral with spectrum , which has period and assume . Let with , . Then the projection onto the span of has the following formula: for ,

 (2.7) (P(λi+pZ)f)(x)=eλi(x)1p∑j∈Zf(x+jp)e−λi(x+jp), for a.e. x∈Ω.
###### Proposition 2.11.

Let be as in Proposition 2.10. For , let . Then for a.e. and, for :

 (2.8) 1p∑j∈Ωxe2πi(λi−λi′)jp=δii′ for a.e. x∈Ω.

In other words , the set is spectral and, for a.e. , is a spectrum for it.

The next theorem exploits the structure of the spectral set described in the previous statements, to explain the form of the group of local translations.

###### Theorem 2.12.

Let be a bounded Borel subset of with . Let . Suppose -tiles by . Then, for a.e. the set

 (2.9) Ωx:={k∈Z:x+kp∈Ω}

has exactly elements

 (2.10) Ωx={k0(x)

For almost every there exist unique and such that .

The functions have the following property

 (2.11) ki(x+1p)=ki(x)−1,(x∈R,i=0,…,p−1).

Consider the space of -periodic vector valued functions . The operator defined by

 (2.12) (Wf)(x)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝f(x+k0(x)p)⋮f(x+kp−1(x)p)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,(x∈[0,1p),f∈L2(Ω)),

is an isometric isomorphism with inverse

 (2.13) W−1⎛⎜ ⎜⎝f0⋮fp−1⎞⎟ ⎟⎠(x)=fi(y), if x=y+ki(y)p, with% y∈[0,1p),i∈{0,…,p−1}.

A set of the form is a spectrum for if and only if is a spectrum for for a.e. .

The exponential functions are mapped by as follows:

 (2.14) (Weλi+np)(x)=eλi+np(x)⎛⎜ ⎜ ⎜ ⎜⎝eλi(k0(x)p)⋮eλi(kp−1(x)p)⎞⎟ ⎟ ⎟ ⎟⎠=:Fi,n(x),(i=0,…,p−1,n∈Z,x∈[0,1p)).

For in define the unitary matrix which has column vectors

 vi(x):=1√p(eλi(k0(x)p),eλi(k1(x)p),…,eλi(kp−1(x)p))t,i=0,…,p−1.

Let be the group of local translations on associated to a spectrum . Consider the one-parameter unitary group on defined by

 (2.15) (Up(t)F)(x)=MxM∗x+tF(x+t),(x,t∈R,F∈L2([0,1p),Cp)).

Then intertwines and :

 (2.16) WUΛ(t)=Up(t)W.
###### Proof.

The first statement follows from the fact that -tiles by . The second statement follows from this and the fact that tiles by .

To check that and , as defined, are inverse to each other requires just a simple computation. We verify that is isometric.

For a subset of with define

 AS:={x∈[0,1p):Ωx=S}.

Note that, since is bounded, for all but finitely many sets . Also we have the following partition of .

 ⋃|S|=p(AS+1pS)=Ω.

Take . We have

 ∥Wf∥2L2([0,1p),Cp)=∫1p0p−1∑j=0∣∣ ∣∣f(x+kj(x)p)∣∣ ∣∣2dx=∑|S|=p∫ASp−1∑j=0∣∣ ∣∣f(x+kj(x)p)∣∣ ∣∣2dx
 =∑|S|=p∫AS∑s∈S∣∣∣f(x+sp)∣∣∣2dx=∑|S|=p∑s∈S∫AS+sp|f(x)|2dx=∫Ω|f(x)|2dx.

Equation (2.14) requires just a simple computation.

If is a spectrum for , then we saw in Proposition 2.11 that has spectrum for a.e. .

For the converse, if is a spectrum for a.e. , then for a.e. ,the vectors , , form an orthonormal basis for . Then the functions in (2.14) form an orthonormal basis for as can be seen by a short computation. To see that the functions span the entire Hilbert space, take in such that for all , . Then

 0=∫1p0eλi+np(x)⟨H(x),vi(x)⟩Cpdx.

Since the functions are complete in it follows that for a.e. . So for a.e. and all . Then for a.e. .

Next, we check that is well defined, so the function in (2.15) is -periodic. We have

 Mx+1p=1√p⎛⎝eλj(ki(x+1p)p)⎞⎠i,j=0,…,p−1=1√p(eλj(ki(x)−1p))i,j,

therefore , where is the diagonal matrix with entries .

We have, for , :

 Mx+1pM∗x+t+1pF(x+t+1p)=MxDλ(1p)∗Dλ(1p)M∗x+tF(x+t)=MxM∗x+tF(x+t).

The fact that the matrices and are unitary implies that is unitary.

To obtain (2.16), it is enough to verify it on the basis and that is equivalent to:

 (2.17) Up(t)Fi,n=eλi+np(t)Fi,n,(i=0,…,p−1,n∈Z).

Note first that for all , . We have

 (Up(t)Fi,n)(x)=MxM∗x+tFi,n(x+t)=√peλi+np(x+t)MxM∗x+tvi(x+t)=√peλi+np(x+t)Mxδi
 =√peλi+np(t)eλi+np(x)vi(x)=eλi+np(t)Fi,n(x).

In the next two propositions we give some equivalent representations of the group of local translations, one involving the usual translation in and the second involving induced representations in the sense of Mackey.

###### Proposition 2.13.

Let be a bounded Borel subset of with . Assume that has a spectrum with period and

 Λ={λ0=0,λ1,…,λp−1}+pZ.

Identify (with the normalized Lebesgue measure), with -periodic -functions. Define the projections as in (2.7), but for all . Define the operator

 (2.18) W:L2(Ω)→L2[0,1p]⊕⋯⊕L2[0,1p]p times ,Wf=(e−λiP(λi+pZ)f)i=1,…,p,(f∈L2(Ω)).

Then is an isometric isomorphism with the following properties:

1. for all , ,

 (2.19) Weλi+pk=(0,…,0,epki-th position,0,…,0)
2. Let be the projection onto the -th component in . Then

 (2.20) WP(λi+pZ)W∗=Pi,i=1,…,p
3. Let be the local translation group associated to . Let be the translation operator on ,

 (2.21) (Ttf)(x)=f(x+t),(x,t∈R).

Then commutes with the projections , and, for ,

 (2.22) WUΛ(t)W∗=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣eλ0(t)Tt0⋯00eλ1(t)Tt⋯0⋱0⋯0eλp−1(t)Tt⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦
###### Proof.

We check (i). Since , (2.19) follows. Since is a spectrum, this implies that maps an orthonormal basis to an orthonormal basis, so it is an isometric isomorphism.

(ii) can be checked on the basis , , . For (iii), we have and . This implies (2.22) and the fact that commutes with the projections .

###### Proposition 2.14.

Let be a bounded Borel subset of with . Assume that has a spectrum with period and

 Λ={λ0=0,λ1,…,λp−1}+pZ.

Let be the one-parameter group of local translations associated to .

Let be a unitary matrix with eigenvalues . Consider the induced representation from to : let be the Hilbert space:

 (2.23) HT:={f:R→Cp:f % measurable ,∫1p0∥f(x)∥2Cpdx<∞,f(x+1p)=Tf(x) for a.e. x∈R},

with inner product

 (2.24) ⟨f,g⟩HT=p∫1p0⟨f(x),g(x)⟩Cpdx.

Define the one-parameter group of unitary transformations

 (2.25) (UT(t)f)(x)=f(x+t)(t,x∈R,f∈HT).

Then there exists a isometric isomorphism from onto that intertwines and , i.e.,

 (2.26) WUΛ(t)=UT(t)W,(t∈R).
###### Proof.

Let be a unitary matrix that diagonalizes , i.e., is the diagonal matrix with entries . Let be the column vectors of , . Note that and form an orthonormal basis for .

Define

 (2.27) Fk,n(x)=eλk+np(x)vk,(x∈R,k=0,…,p−1,n∈Z).

We check that is in . For every ,

 Fk,n(x+1p)=eλk+np(x+1p)vk=eλk+np(x)eλk(1p)vk=eλk+np(x)Tvk=TFk,n(x).

Thus, .

We prove that is an orthonormal basis for . We have

 ⟨Fk,n,Fl,m⟩HT=p∫1p0⟨vk,vl⟩Cp¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯eλk+np(x)eλl+mp(x)dx=δklp∫1p0¯¯¯¯¯¯¯¯¯¯¯¯¯¯enp(x)emp(x)dx=δklδmn.

So are orthonormal in . We prove that they are complete. Let such that for all , . Then

 0=⟨H,Fk,n⟩HT=p∫1p0⟨H(x),eλk(x)vk⟩Cpenp(x)dx.

Then, since form an ONB in we get that for a.e. .

But, note that

so the function is -periodic. Therefore for a.e. .

Since form an ONB in , we get that for a.e. . Thus is complete.

Define from onto by for all , . Then extends linearly to an isometric isomorphism.

We have, for , , , :

 (WUΛ(t)eλk+np)(x)=(W(eλk+np(t)eλk+np))(x)=eλk+np(t)Fk,n(x)=eλk+np(t)eλk+np(x)vk
 =eλk+np(x+t)vk=Fk,n(x+t)=(