A Constructive Proof of Beurling-Lax Theorem∗

Qiuhui CHEN Tao QIAN

1 Introduction

In 1948,Beurling[2] first investigated the shift-operator-invariant subspace M in Hardy spaceH2(D)in the unit disc context for the shift operator defined by

Here a subspace M inH2(D)is said to be a shift-operator-invariant-subspace if

The following celebrated result is called Beurling theorem.

Theorem 1.1Suppose thatMis a non-zero closed subspace of H2(D).ThenMis a shift-operator-invariant space if and only if

where I is an inner function in H∞(D).

Here we say a functionis an inner function if|f|=1 almost everywhere on the boundary∂D.Similarly,a functionis an inner function if|f|=1 almost everywhere on the real line R.

As a consequence of this theorem,a subspace is a backward-shift-invariant space(SM⊂M)if and only if(see[3])

where the backward-shift operator is defined by

The Beurling theorem was extended to the upper-half complex plane by Lax in 1959,which is now named the Beurling-Lax theorem(see[7]).In the upper-half plane case,by using the same notation,the shift operator and the backward-shift operator are defined,respectively,by

Theorem 1.2LetMbe a non-zero closed subspace of H2(C+).ThenMis a shift-operatorinvariant subspace if and only if there exists an inner function F∈H∞(C+)such thatM=FH2(C+).

There is a large number of documents for the studies of forward shift and backward-shift invariant subspaces(see[1,3–5])mainly in theoretical aspects.As an important tool,the shift invariant subspace is useful in bandwidth keeping and phase retrieval,which are two topics in optical and signal processing(see[6]).Recently,Tan and Qian used techniques of forward and backward operators to solve some questions in the theory of analytic signals,especially the characterization of Bedrosian identity.We will offer a survey in Section 2 about Tan and Qian’s results.We are inspired to revisit the proof of the Beurling-Lax theorem.

We remark that,in the unit disc case,Beurling’s proof is constructive for the choice of an inner function for any invariant subspace,which is important for the calculation and characterization of analytic signals from the Bedrosian identity.But in the upper half plane case,the proof of the Beurling-Lax theorem(see[5,7])is complicated by involving the technique of isomorphism betweenH2(D)andH2(C+).Many applications are dependent on the concrete form of the factor inner function inducing the shift invariant subspace.In this note,we choose the special functionand use its orthogonal projection into M to construct the inner functionFappearing in Theorem 1.2.

2 Preliminaries and Surveys

Related to the Hardy space in the unit disc,we sayf∈Hp(D),if

forp∈(0,∞).Whenp=∞,we sayis a bounded analytic function on D and we write

In the upper half plane case,we sayis analytic on C+and

Whenp=∞,we writef∈H∞(C+)for the bounded analytic functions on C+,and we givethe norm

In the casep=2,as Hilbert spaces,H2(D)andH2(C+)are equipped with the inner product

and

respectively.

The linear mapping which takes a functionto its non-tangential boundary limit functionf∈Lp(R)is an isometric isomorphism fromHp(C+)onto a closed subspace ofLp(R)which we shall denote byHp(R).

For any functionby the well-known Nevanlinna factorization theorem,fcan be factorized as

where

(1)Of,the outer factor off,is given by

(2)If,the inner factor off,can be further factorized into

where

(i)bis a real number andais a nonnegative number;

(ii)Bfis the Blaschke product formed with the zero sequenceoffin C+(the zeros repeat according to their respective multiples)which can be represented by

whereis understood whenever

(iii),the singular inner function,is given by

whereμis a real,bounded,increasing function with derivativeμ?(t)=0 almost everywhere on R.

Two classical problems of long interest in a number of practical areas,including optics,antenna theory and physics,are formulated as follows:The first is to find all functionsgsuch that Band(fg)⊂Band(f);the second is to find all-pass filters eiθ(t)such that Band(feiθ)⊂Band(f).Here,Band(f)is the support of the bandlimited signalf∈L2(R)whose Fourier transform has compact support.The Lebesgue measure of Band(f)is called the bandwidth off.They are referred to as the band keeping problem and the phase retrieval problem,respectively.Obviously,the phase retrieval problem is closely related to the band keeping problem.In[10],the authors give a characterization of functiongsuch that Band(fg)⊂Band(f).

Proposition 2.1Suppose that f,g are nonzero functions,f∈L2(R)is bandlimited withBand(f)=[0,A]andThen fg∈L2(R)is bandlimited withBand(fg)=[0,A]if and only ifwhere Ifis the inner factor of f which does not contain the singular inner function.

Noting thatis a backward-shift invariant subspace,the authors in[10]offer a characterization of this subspace.

Proposition 2.2Suppose thatis a sequence inC+which can define a Blaschke product

Then for p∈(1,∞),

where the closureis in the Lptopology,:?f,Bg?=0,∀g∈Hp?}and the Takenaka-Malmquist(TM,for short)system defined by

Moreover,in[8],the authors prove the following result.

Proposition 2.3The TM system{en}is a Schauder basis inIn other words,for any we have

It consequently concludes that the TM system{en}is a Schauder basis ofHp(R)if{αk}can not form a Blaschke product.

3 A Constructive Proof of Lax’s Theorem

The sufficiency is trivial sinceFH2(C+)is an invariant subspace under the multiplication by eiλw.Now we show the necessity.Suppose that M is a non-zero closed subspace ofLetbe the orthogonal projection of the functiononto M.Therefore,

whereG∈M andw+iis orthogonal to M.

Now we prove thatfort∈R,whereCis some positive constant.SinceGis in M,we know thatis in M for anydue to the invariance of M under the multiplication by eiλw.By the orthogonality between M and,we have

It follows that

Since eiλzG(z)is analytic on the upper half plane whenλ>0,we apply Cauchy’s formula to get that

SinceG∈H2(C+),by Nevanlinna factorization theorem,we havewithIG∈andbeing inner function and outer function,respectively.Now we define the subspaceby

Obviously,is an invariant subspace under the multiplication by eiλwfor anyλ≥0.We observe thatGis also insinceBy using the density ofinand the invariance of M under multiplication by eiλw,we conclude that

Finally,we want to show thatTo this end,it suffices to show that the conditionsf∈M andimply thatf=0.On the one hand,from the invariance ofunder multiplication by eiλwandwe have

On the other hand,from the facts thatand the invariance of M under the multiplication by eiλw,we obtain that

which is equivalent to

Settingλ=0 in Equations(3.1)–(3.2),we conclude thatf(i)=0.Therefore,(3.1)–(3.2)indicate that the Fourier transform ofis zero.ThusNoting that we have already shownwe therefore concludef=0.The proof of this theorem is completed.

AcknowledgementThe authors are grateful to the anonymous referees for their helpful comments and suggestions.

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