An F. and M. Riesz Theorem for strong H2-functions on the unit circle

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Abstract

An F. and M. Riesz Theorem states that if f,gH2 are such that fg=0 a.e. on the unit circle T then either f=0 a.e. on T or g=0 a.e. on T. In this paper we establish an extension of the F. and M. Riesz Theorem for (operator-valued) strong H2-functions on the unit circle.

Introduction

Let H2H2(T) be the Hardy space of the unit circle T in the complex plane C. An F. and M. Riesz Theorem states that if f,gH2, thenfg=0a.e.either f=0 a.e. or g=0 a.e.. (cf. [4]). However, the statement (1) is liable to fail for operator-valued functions because they have zero divisors even though f and g are matrix-valued functions. Thus we may ask under what condition on operator-valued functions f and g does the statement (1) hold? In this paper, we extend the statement (1) for operator-valued functions.

We first review a few essential facts concerning vector-valued Lp- and Hp-functions, using [3], [6], [7], [8], and [10] for general references. For a complex Banach space X and 1p<, let LXpLp(T,X) be the set of all strongly (Lebesgue) measurable functions f:TX such that||f||p:=(12πππ||f(eit)||Xpdt)1p<. For fLX1, the n-th Fourier coefficient of f, denoted by fˆ(n), is defined byfˆ(n):=12πππeintf(eit)dtfor each nZ, where the integral is understood in the sense of the Bochner integral. For 1p<, defineHXp:={fLXp:fˆ(n)=0forn<0}. If E is a complex Hilbert space then LE2 is a complex Hilbert space with the inner product defined byf,gLE2:=12πππf(eit),g(eit)Edt. Let D be the open unit disk in C. For a complex Banach space X, denote H2(D,X) for the set of all functions analytic on D satisfying||f||H2(D,X):=sup0<r<1(12πππ||f(reit)||X2dt)12<. Throughout the paper, suppose D and E are infinite-dimensional separable complex Hilbert spaces. As in the scalar-valued case, if fH2(D,E), then there exists a “boundary function” bfHE2 such thatf(reiθ)=P[bf](reiθ)(r[0,1) and θR), where P[] denotes the Poisson integral. The Taylor coefficients of f coincide with the Fourier coefficients of bf, and (bf)(eiθ)=limr1f(reiθ) a.e. Moreover, the mapping fbf is an isometric bijection from H2(D,E) onto HE2 (cf. [6, Theorem 3.11.7]). Thus if gHE2, then there exists fH2(D,E) such that bf=g. In this case, we write fP[g].

Recall that a function Φ:DB(D,E) is called a strong H2-function if Φ()xH2(D,E) for each xD (cf. [5]). Write Hs2(D,B(D,E)) for the space of B(D,E)-valued strong H2-functions on D. On the other hand, for 1p<, define the class Lsp(T,B(D,E)) by the set of all WOT measurable B(D,E)-valued functions Φ on T satisfyingΦ()xLEpfor every xD. Following to V. Peller [7], a function ΦLsp(T,B(D,E)) is called a strong Lp-function.

For ΦLs2(T,B(D,E)), a Toeplitz operator TΦ:HD2HE2 is a densely defined operator defined byTΦp:=P+(Φp)(pPD), where P+ denotes the orthogonal projection from LE2 onto HE2 and PD is the set of all polynomials with values in D. Very recently, a systematic study on strong L2-functions was undertaken in the study of SE-invariant subspaces (see [2]), where SE is a shift operator on HE2 defined by (SEf)(z):=zf(z) for each fHE2. If ΦLs1(T,B(D,E)) and xD, then Φ()xLE1. We now define the n-th Fourier coefficient of ΦLs1(T,B(D,E)), denoted by Φˆ(n), byΦˆ(n)x:=Φ()xˆ(n)(nZxD). It is easy to show that Φˆ(n) is linear transformation from D to E for each nZ. We defineHs2(T,B(D,E)):={ΦLs2(T,B(D,E)):Φˆ(n)=0forn<0}, or equivalently, Hs2(T,B(D,E)) is the set of all WOT measurable functions Φ on T such that Φ()xHE2 for each xD. A function AHs2(T,B(D,E)) is called a strong H2-function on the unit circle T. We can easily check thatHB(D,E)2Hs2(T,B(D,E)).

In this paer, we establish an extension of the statement (1) for strong H2-functions on the unit circle T.

For ΦLs2(T,B(D,E)) and αD, writeBα(z):=zα1αz(a Blaschke factor)andΦα:=ΦBα.

Our main theorem now states:

Theorem 1.1

Let AHs2(T,B(D,E)) and CHs2(T,B(D,D)). Suppose the 0-th Fourier coefficient Cαˆ(0) of Cα has dense range for some αD. If AC=0 a.e. on T, then A=0 a.e. on T.

Let Mm×n denote the set of m×n complex matrices and write Hm×n2 for the set of m×n matrix-valued H2-functions, i.e.,Hm×n2=H2Mm×n. We can easily check that Hm×n2=Hs2(T,B(Cn,Cm))=H2(D,Mm×n).

As a corollary of Theorem 1.1, we then have:

Corollary 1.2

Let AHm×n2 and CHn×r2. Suppose A(α) or C(α) is injective for some αD. If AC=0 a.e. on T, then either A=0 a.e. on T or C=0 a.e. on T.

A significance of the Corollary 1.2 lies in the fact that some condition on the front factor or the back factor of the product holds for at least one point in the open unit disk then an operator-valued version of the F. and M. Riesz Theorem (1) holds.

To illustrate Corollary 1.2, considerC=[BαfgBα](αD,f,gH2withf(α)0g(α)). Clearly, CH2×22. ObserveC(α)=[0f(α)g(α)0] is invertible. Thus by Corollary 1.2, AC0 for any nonzero AHm×22 (mZ+).

In Section 2, we give proofs of Theorem 1.1 and Corollary 1.2.

Section snippets

Proofs of the main results

To prove Theorem 1.1 and Corollary 1.2, we need several auxiliary lemmas. We begin with recalling the Poisson kernel defined byPr(t):=n=r|n|eint=1r212rcost+r2(0r<1,tR). If fLX1 for a complex Banach space X, let P[f] denote the Poisson integral of f defined byP[f](reiθ):=12πππPr(θt)f(eit)dt. For α=reiθD, let bα(t):=Bα(eit) for tR. Then we can see thatbα(t)=|bα(t)|=Pr(θt)andbα(t)=1+r1r<. Thus bα maps sets of measure 0 to sets of measure 0 and in turn, bα maps measurable sets

Acknowledgments

The authors are indebted to the referee for many helpful suggestions. The work of the second named author was supported by NRF (Korea) grant No. 2019R1A2C1005182. The work of the third named author was supported by NRF (Korea) grant No. 2019R1A6A1A10073079. The work of the fourth named author was supported by NRF (Korea) grant No. 2018R1A2B6004116.

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