Doubly invariant subspaces of the Besicovitch space

A classical result of Norbert Wiener characterises doubly shift-invariant subspaces for square integrable functions on the unit circle with respect to a finite positive Borel measure $\mu$, as being the ranges of the multiplication maps corresponding to the characteristic functions of $\mu$-measurable subsets of the unit circle. An analogue of this result is given for the Besicovitch Hilbert space of `square integrable almost periodic functions'.


Introduction
The aim of this article is to prove a version of the classical result due to N. Wiener, characterising doubly shift-invariant subspaces (of the Hilbert space square integrable functions on the circle with respect to a finite positive Borel measure), for the Besicovitch Hilbert space.We give the pertinent definitions below.
The multiplication operator M z : L 2 µ pTq Ñ L 2 µ pTq is given by pM z f qpwq " wf pwq for all w P T, f P L 2 µ pTq, and is called the shift-operator.
The following result gives a characterisation of doubly invariant subspaces of L 2 µ pTq: µ pTq be a closed subspace of L 2 µ pTq.Then M z E " E if and only if there exists a unique measurable set σ Ă T such that E " 1 σ L 2 µ pTq " tf P L 2 µ pTq : f " 0 µ-a.e. on T zσu.We will prove a similar result when L 2 µ pTq is replaced by AP 2 , the Besicovitch Hilbert space.We recall this space and a few of its properties in the following section, before stating and proving our main result in the final section.

Preliminaries on the Besicovitch space AP 2
For λ P R, let e λ :" e iλ¨P L 8 pRq.Let T be the space of trigonometric polynomials, i.e., T is the linear span of te λ : λ P Ru.The Besicovitch space AP 2 is the completion of T with respect to the inner product ´R ppxq qpxq dx, for p, q P T , and where ¨denotes complex conjugation.We remark that elements of AP 2 are not to be thought of as functions on R: For example, consider the sequence pq n q in T , where Then pq n q converges to an element of AP 2 , but pq n pxqq nPN diverges for all x P R (see [6,Remark 5.1.2,p.91]).Although elements of AP 2 may not be functions on R, they can be identified as functions on the Bohr compactifcation R B of R, and we elaborate on this below.We refer the reader to [1, §7.1] and the references therein for further details.
For a locally compact Abelian group G written additively, the dual group G is the set all continuous characters of G. Recall that a character of G is a map χ : G Ñ T such that χpg `hq " χpgq χphq pg, h P Gq.
Then G ˚becomes an Abelian group with pointwise multplication, but we continue to write the group operation in G ˚also additively, motivated by the special characters when G " R.So the inverse of χ P G ˚is denoted by ´χ.Then G ˚is a locally compact Abelian group with the topology given by the basis formed by the sets U g 1 ,¨¨¨,gn;ǫ pχq :" tη P G ˚: |ηpg i q ´χpg i q| ă ǫ for all 1 ď i ď nu, where ǫ ą 0, n P N :" t1, 2, 3, ¨¨¨u, g 1 , ¨¨¨, g n P G.
Let G d denote the group G ˚with the discrete topology.The dual group pG d q ˚of G d is called the Bohr compactification of G.By the Pontryagin duality theorem1 , G is the set of all continuous characters of G ˚, and since G B is the set of all (continuous or not) characters of G ˚, G can be considered to be contained in G B .It can be shown that G is dense in G B .Let µ be the normalised Haar measure in G B , that is, µ is a positive regular Borel measure such that ‚ (invariance) µpUq " µpU `ξq for all Borel sets U Ă G B , and all ξ P G B , ‚ (normalisation) µpG B q " 1.Let R B " pR d q ˚denote the Bohr compactification of R. Let µ be the normalised Haar measure on R B .Let L 2 µ pR B q be the Hilbert space of all functions with pointwise operations and the inner product f pξq gpξqdµpξq for all f, g P L 2 µ pR B q.The Besicovitch space AP 2 can be identified as a Hilbert space with L 2 µ pR B q, and let ι : AP 2 Ñ L 2 µ pR B q be the Hilbert space isomorphism.Let L 8 µ pR B q be the space of µ B -measurable functions that are essentially bounded (that is bounded on R B up to a set of measure 0) with the essential supremum norm }f } 8 :" inftM ě 0 : |f pξq| ď M a.e.u.
For an element f P L 8 µ pR B q, let M f : L 2 µ pR B q Ñ L 2 µ pR B q be the multiplication map ϕ Þ Ñ f ϕ, where f ϕ is the pointwise multiplication of f and ϕ as functions on R B .
Let AP Ă L 8 pRq be the C ˚-algebra of almost periodic functions, namely the closure in L 8 pRq of the space T of trigonometric polynomials.Then it can be shown that AP Ă AP 2 , and ιpAP q " CpR B q Ă L 8 µ pR B q. Also, for For f, g P AP , and λ P R, ιpf gq " pιf qpιgq, ιpe 0 q " 1 R B , ιpe λ q " ιpe λ q " ιpe ´λq.
Every element f P AP gives a multiplication map, M ιpf q on L 2 µ pR B q.For f P AP 2 , the mean value pιpf qqpξq dµpξq " xιf, ι e 0 y " xf, e 0 y exists.The set Σpf q :" tλ P R : mpe ´λf q ‰ 0u is called the Bohr spectrum of f , and can be shown to be at most countable.We have a Hilbert space isomorphism, via the Fourier transform, between L 2 pTq and ℓ 2 pZq: Analogously, we have a representation of AP 2 via the Bohr transform.We elaborate on this below.Let ℓ 2 pRq be the set of all f : R Ñ C for which the set tλ P C : f pλq ‰ 0u is countable and }f } 2 2 :" Then ℓ 2 pRq is a Hilbert space with pointwise operations and the inner product xf, gy " ÿ λ P R f pλq gpλq.
Let c 00 pRq Ă ℓ 2 pRq be the set of finitely supported functions.Define the map F : c 00 pRq Ñ AP 2 as follows: For f P c 00 pRq, pF f qpxq " f pλq e iλx px P Rq.
By continuity, F : c 00 pRq Ñ AP 2 can be extended to a map (denoted by the same symbol) F : ℓ 2 pRq Ñ AP 2 defined on all of ℓ 2 pRq, and is called the Bohr transform.The map F : ℓ 2 pRq Ñ AP 2 is a Hilbert space isomorphism.The inverse Bohr transform F ´1 : AP 2 Ñ ℓ 2 pRq is given by pF ´1f qpλq " mpf e ´λq pλ P Rq.
For λ P R and f P L 2 µ pR B q, we have the following equality in ℓ 2 pRq: F ´1ι ´1pM ιpe λ q f q " pF ´1ι ´1f qp¨´λq " S λ pF ´1ι ´1f q.
We also note that by the Cauchy-Schwarz inequality in L 2 µ pR B q, for all functions f, g P L 2 µ pR B q, we have " }f } 2 2 }g} 2 2 .We will need the following approximation result (see e.g.[2] or [1]): Proposition 2.1.Let f P AP and Σpf q be its Bohr spectrum.Then there exists a sequence pp n q nPN in T such that ‚ for all n P N, Σpp n q Ă Σpf q, and ‚ pp n q nPN converges uniformly to f on R.
Analogous to the classical Fourier theory where the Fourier coefficients of the pointwise multiplication of sufficiently regular functions is given by the convolution of their Fourier coefficients, we have the following.
Now consider the general case when f P L 8 µ pR B q and g P L 2 µ pR B q. Then we can find sequences pf n q, pg n q in ιT such that ‚ pf n q nPN converges uniformly to f , ‚ pg n q nPN converges to g in AP 2 , and ‚ for all n P N, Σpι ´1f n q Ă Σpι ´1f q, and Σpι ´1g n q Ă Σpι ´1gq.We remark that the g n can be constructed by simply 'truncating' the 'Bohr series' of ι ´1g, since sup Then, with p ¨:" F ´1ι ´1, we have ˇˇÿ This completes the proof.

Characterisation of doubly invariant subspaces
In this section, we state and prove our main results, namely Theorem 3.1 and Corollary 3.2.Theorem 3.1 is a straightforward adaptation2 of the proof of the classical version of the theorem given in [5, Theorem 1.2.1, p.8].On the other hand, the main result of the article is Corollary 3.2, which follows from Theorem 3.1 by an application of Lemma 2.2.For a measurable set σ Ă R B , let 1 σ P L 8 µ pR B q denote the characteristic function of σ, i.e., Theorem 3.1.Let E Ă L 2 µ pR B q be a closed subspace of L 2 µ pR B q. Then the following are equivalent: (1) M ιpe λ q E " E for all λ P R.
(1)ñ(2): Let P E : L 2 µ pR B q Ñ L 2 µ pR B q be the orthogonal projection onto the closed subspace E. Set f " P E 1 R B .Let I be the identity map on L 2 µ pR B q.We claim that 1 R B ´f K E. p‹q Indeed, for all g P E, " xP E pI ´PE q1 R B , gy " x0, gy " 0.
As f " P E 1 R B P E and M ιpe λ q E " E for all λ P R, we have 1 R B ´f K M ιpe λ q f for all λ P R.So for all p P T , ż But T is dense in AP 2 , and µ is a finite positive Borel measure.So f p1 R B ´f q " 0 µ-a.e.
Then f " 1 σ µ-a.e.As 1 σ " f " P E 1 R B P E, and as M ιpe λ q E " E for all λ P R, it follows that 1 σ ιpT q Ă E. But E is closed, and thus closurep1 σ ιpT qq Ă E.
Since closurepT q " AP 2 , we conclude that 1 σ L 2 µ pR B q Ă E. Next we want to show that E Ă 1 σ L 2 µ pR B q. To this end, let g P E be orthogonal to 1 σ L 2 µ pR B q.In particular, for all λ P R, ż R B g 1 σ ιpe λ q dµ " 0. p˚q We want to show that g " 0. Since g P E, M ιpe λ q g P E for all λ.So by (‹) above, 1 R B ´1σ K M ιpe λ q g, and noting that 1 R B , 1 σ are real-valued, ż R B g ιpe λ qp1 R B ´1σ q dµ " 0. p˚˚q Hence, using the density of ιpT q in L 2 µ pR B q, we obtain from (˚) and (˚˚) that g 1 σ " 0 µ-a.e.g p1 R B ´1σ q " 0 µ-a.e.
The uniqueness of σ up to a set of µ-measure 0 can be seen as follows: If E " M 1σ L 2 µ pR B q " M 1 σ 1 L 2 µ pR B q, then taking 1 B P L 2 µ pR B q, we must have 1 σ " 1 σ 1 ϕ for some ϕ P L 2 µ pR B q.So σ Ă σ 1 .Similarly, σ Ă σ 1 as well.We now interpret the above characterisation result for doubly invariant subspaces of AP 2 in terms of the Bohr coefficients of elements of E. Given a measurable set σ Ă R B , define p σ P ℓ 2 pRq by Then the following are equivalent: (1) S λ E " E for all λ P R.