Bases of reproducing kernels in model spaces

Description

Niveau: Supérieur, Doctorat, Bac+8
Bases of reproducing kernels in model spaces EMMANUEL FRICAIN Abstract This paper deals with geometric properties of sequences of reproducing kernels related to invariant subspaces of the backward shift operator in the Hardy space H2. Let ? = (?n)n>1 ? D, ? be an inner function in H∞(L(E)), where E is a finite dimensional Hilbert space, and (en)n>1 a sequence of vectors in E. Then we give a criterion for the vector valued reproducing kernels (k?(., ?n)en)n>1 to be a Riesz basis for K? := H2(E) ?H2(E). Using this criterion, we extend to the vector valued case some basic facts that are well-known for the scalar valued reproducing kernels. Moreover, we study the stability problem, that is, given a Riesz basis (k?(., ?n))n>1, we characterize its perturbations (k?(., µn))n>1 that preserve the Riesz basis property. For the case of asymptotically orthonormal sequences, we give an effective upper bound for uniform perturbations preserving stability and compare our result with Kadecˇ's 1/4-theorem. 1 Introduction This paper is devoted to geometric properties of sequences of reproducing kernels in Hardy spaces. These properties are of interest for several reasons.

• stability properties


• operator ?


• vector valued


• reproducing kernels


• perturbation


• perturbations preserving stability


• hilbert space


• valued reproducing


• all ? ?

Sujets

• Perturbation
• Hilbert space

Informations

• Publié par: mijec
• Nombre de lectures: 33
• Langue: English

Extrait

Bases

of

reproducing kernels spaces

EMMANUEL FRICAIN

Abstract

in

model

This paper deals with geometric properties of sequences of reproducing kernels related to invariant subspaces of the backward shift operator in the Hardy space H 2  Λ = (. Letλ n ) n>1 ⊂D, Θ be an inner function inH ∞ (L(E)), whereEis a finite dimensional Hilbert space, and (e n ) n>1 a sequence of vectors inE. Then we give a criterion for the vector valued reproducing kernels (k Θ (., λ n )e n ) n>1 to be a Riesz basis forK Θ :=H 2 (E)ΘH 2 (E). Using this criterion, we extend to the vector valued case some basic facts that are well-known for the scalar valued reproducing kernels. Moreover, we study the stability problem, that is, given a Riesz basis (k Θ (., λ n )) n>1 , we characterize its perturbations (k Θ (., µ n )) n>1 that preserve the Riesz basis property. For the case of asymptotically orthonormal sequences, we give an effective upper bound for uniform perturbations preserving stability and compareourresultwithKadeˇc’s1/4-theorem.

1 Introduction

This paper is devoted to geometric properties of sequences of reproducing kernels in Hardy spaces. These properties are of interest for several reasons. First of all, if we consider aC 00 contraction,-Tthen by Sz-Nagy and C. Foias’, in a Hilbert space, theory,Tis unitarily equivalent to its canonical modelM Θ , M Θ f:=P Θ (zf), f∈K Θ . HereK Θ is the model space

K Θ :=H 2 (E)ΘH 2 (E), Eis an auxiliary Hilbert space,H 2 (E) stands for theE-valued Hardy space in the unit discD:={z∈C:|z|<1}, Θ is the characteristic function ofTandP Θ

1

is the orthogonal projection ontoK Θ  spectral theory of. TheM Θ in the language of the characteristic function Θ (see [23], [18], [20]) depends on the geometry of reproducing kernels ofK Θ , that is, on the (operator-valued) functions k(z, λ ) 1−Θ(z)Θ(λ) ∗ , z, λ∈D, Θ := 1−λz

satisfying hf(λ), ei=hf, k Θ (., λ)ei H 2 (E) , for everyλ∈D,e∈Eandf∈K Θ  in some approaches, the kernels. Moreover, k Θ (z, λ) and their various analogs are the starting point for the model theory and its applications, especially to various interpolation problems; see [7], [3] and [10]. Our goal in this paper is to extend to the vector valued case some basic facts that are well-known for the scalar valued reproducing kernels (E=C, see [18], [12]). Recall that a scalar inner functionϑcan be written asϑ=SB, whereSis the singular inner factor ofϑ, z dµ(ζ S(z) = exp  − Z T ζ ζ+ −z )  ,

the measuredµbeing positive and singular with respect to the Lebesgue measure dm, andBis the corresponding Blaschke product, B= Y b λ n , n>1 b λ n := | λ λ n n | 1 λ − n λ − n z z . It is known (see [18]) that z− ϑ λ n ∈K ϑ are eigenvectors ofM ϑ , and 1 ϑ are eige k λ n := 1−λ n z ∈Knvectors of the adjoint operatorM ϑ ∗ : M ϑ  z−λ ϑ n  =λ n z− ϑ λ n , M ϑ ∗ k λ n =λ n k λ n . Moreover, it is proved in [12] that the union of eigenvectors ofM ϑ andM ϑ ∗ forms an unconditional basis (a Riesz basis) inK ϑ if and only if the reproducing kernels (k S (., λ n )) n>1 form the same kind of basis inK S  latter property is, therefore,. The important for some applications such as the string scattering theory, see [12] and [17] for details. Secondly, some motivation comes from control theory and signal processing. Namely, in the special case where Θ = Θ a := exp(a z z −1 1+ ), the reproducing kernels k Θ (., λ), withλ∈D, arise as images of the exponential functions exp(−iµω)χ (0,a) , withµ:=i 1 1− + λ λ , under a natural unitary map going fromL 2 (0, a) toK Θ . The prop- erty of families of exponentials (and hence, of reproducing kernelsk Θ for Θ = Θ a )

2

to be a Riesz basis is important for control theory. Indeed, under some further hypotheses, the theory of controllability of a dynamic systemx 0 (t) =Ax(t) +Bu(t), t>0,x(0) =x 0 , depends on the geometric properties of the system of exponentials (exp(−λ n t)B ∗ ψ n ) n>1 , where (ψ n ) n>1 is the sequence of eigenvectors ofA ∗ , associ- ated to the sequence of eigenvalues (λ n ) n>1  more details on the relationships. For between the controllability problems and the geometry of families of exponentials we refer to [1] and [19].

A third reason to study geometric properties of sequences of reproducing kernels is just to understand better non-harmonic exponentials (exp(iλ n t)) n>1 which appear frequently, say, in analysis of convolution equations (see, for instance, [12] and [29]). To solve the problem of Riesz bases for the families (k Θ (., λ n )) n>1 in scalarK Θ , S.V. Hruschev, N.K. Nikolski and B.S. Pavlov proposed in [12] a new method (see also [18]). They gave a criterion in terms of the Carleson condition and invertibility of a certain Toeplitz operator. In the first part of this paper, using the above- mentioned operator approach, we give vector valued generalizations of several results of Hruschev, Nikolski and Pavlov.

Another subject of this paper concerns stability properties of reproducing kernel bases (k Θ (., λ n )) n>1 under small perturbations of the polesλ n . A reason to study the stability properties is that the criteria mentioned above involve, however, some properties of a given family (k Θ (., λ n )e n ) n>1 that are rather difficult to verify. On the other hand, in many cases, the given family is a slight perturbation of another family (k Θ (., µ n )e 0 n ) n>1  gives rise to the stabil-that is known to be a basis. This ity problem whose formal statement can be found in section 3.1 below. Roughly speaking, given a Riesz basis (k Θ (., λ n )) n>1 inK Θ , the problem is to characterize its perturbations (k Θ (., µ n )) n>1 that still enjoy the property of being a Riesz basis. In fact, this problem was initially raised by R. E. A. C. Paley and N. Wiener for the orthogonal basis (exp(int), n∈Z) inL 2 (0,2π sufficient condition for such). A stability is given in [21]. For this case, the problem of uniform stability (see section 3.1)wascompletelysolvedbyA.InghamandM.Kadecˇ(see[15]and[14]).In1990, A. Avdonin and I. Joo gave a sufficient condition for stability of general uncondi- tional bases of exponentials. In section 3, we give a generalization of this result for the reproducing kernels (k Θ (., λ n )e n :n>1). The paper is organized as follows. In section 2, we deal with Riesz bases of vector valued reproducing kernels (k Θ (., λ n )e n ) n>1  goal is to separate, as far as it. Our is possible, the influence of the three parameters involved: the frequency spectrum Λ = (λ n ) n>1 , the spatial directionse n ,n>1, and the inner function Θ generating the model subspaceK Θ . In particular, ifEis a finite dimensional Hilbert space, we prove, in Theorem 2.1, the vector analog of Hruschev, Nikolski and Pavlov’s criterion for a family (k Θ (., λ n )e n ) n>1 to be a Riesz basis forK Θ . If (k λ n e n ) n>1 is not a Riesz basis forK B , then we adapt a method proposed by V. Vasyunin in

3