Bases of reproducing kernels in de Branges spaces E Fricain

Girard Desargues

Description

Bases of reproducing kernels in de-Branges spaces E. Fricain? August 18, 2004 Abstract This paper deals with geometric properties of sequences of reproducing kernels related to de-Branges spaces. If b is a nonconstant function in the unit ball of H∞, and Tb is the Toeplitz operator, with symbol b, then the de- Branges space, H(b), associated to b, is defined byH(b) = (Id?TbTb)1/2H2, where H2 is the Hardy space of the unit disk. It is equiped with the inner product such that (Id? TbTb)1/2 is a partial isometry from H2 onto H(b). First, following a work of Ahern-Clark, we study the problem of orthogonal basis of reproducing kernels in H(b). Then we give a criterion for sequences of reproducing kernels which form an unconditionnal basis in their closed linear span. As far as concerns the problem of complete unconditionnal basis in H(b), we show that there is a dichotomy between the case where b is an extreme point of the unit ball of H∞ and the opposite case. Keywords: de-Branges spaces, Riesz bases, reproducing kernels. 2000 ams subject classification: 46E22, 47B32, 47B38, 30H05, 46J15, 46B15. Acknowledgements.

• inner product


• nontrivial invariant


• invariant subspace


• geometric properties


• branges space


• facts concerning


• problems such


• positive borel


• subspace bh2

Sujets

• Desargues
• Inner product space
• Invariant subspace
• Geometry

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 15
• Langue: English

Extrait

Bases of reproducing kernels in de-Branges spaces
∗E. Fricain
August 18, 2004
Abstract
This paper deals with geometric properties of sequences of reproducing
kernels related to de-Branges spaces. If b is a nonconstant function in the
∞unitballofH ,andT istheToeplitzoperator,withsymbolb,thenthede-b
1/2 2Brangesspace,H(b),associatedtob,isdefinedbyH(b) = (Id−T T ) H ,b b
2where H is the Hardy space of the unit disk. It is equiped with the inner
1/2 2product such that (Id−T T ) is a partial isometry from H ontoH(b).b b
First,followingaworkofAhern-Clark,westudytheproblemoforthogonal
basisofreproducingkernelsinH(b). Thenwegiveacriterionforsequences
of reproducing kernels which form an unconditionnal basis in their closed
linear span. As far as concerns the problem of complete unconditionnal
basis in H(b), we show that there is a dichotomy between the case where
∞b is an extreme point of the unit ball of H and the opposite case.
Keywords: de-Branges spaces, Riesz bases, reproducing kernels.
2000 ams subject classification: 46E22, 47B32, 47B38, 30H05, 46J15, 46B15.
Acknowledgements. This work was done while the author was visiting the
Department of Mathematics and Statistics of the University of Laval (Quebec).
He is grateful to the Department members, specially to T. Ransford, for their
warm hospitality.
∗Institut Girard Desargues, UFR de Math´ematiques, Universit´e Claude Bernard Lyon 1,
69622 Villeurbanne Cedex, France. [e-mail].
11 Introduction
This paperis devoted to geometric properties of sequences of reproducing kernels
inde-Brangesspaces. Thesespaces,firststudiedbyL.deBrangesandJ.Rovnyak
2[6], are (not necessarily closed) subspaces of the Hardy space H of the unit disk,
. Recall first that
Z
2 2H := f : → analytic : sup |f(rζ)| dm(ζ)<∞ ,
06r<1
where is the unit circle and dm is the normalized Lebesgue measure on .
2 2 2As usual, H will be identified (via radial limits) with the space of L = L ( )
functions whose negatively indexed Fourier coefficients vanish. Norm and inner
2 2product in L or H will be denoted byk·k andh·,·i , respectively.2 2
2 2 ∞Let P denote the orthogonal projection of L onto H . For ϕ∈ L , let T+ ϕ
2denote the Toeplitz operator with symbol ϕ defined on H by T f = P (ϕf).ϕ +
2Thede-Brangesspace,H(ϕ),associatedtoϕconsistsofthoseH functionswhich
1/2belong to the range of the operator (Id−T T ) . It is a Hilbert space whenϕ ϕ
equipped with the inner product
hf,gi :=hP ⊥f ,P ⊥g i ,ϕ Ker(Id−T T ) 1 Ker(Id−T T ) 1 2ϕ ϕϕ ϕ
1/2 1/2where f = (Id−T T ) f , g = (Id−T T ) g and P ⊥ denotes theϕ ϕ 1 ϕ ϕ 1 Ker(Id−T T )ϕ ϕ
2 ⊥orthogonal projection of H onto Ker(Id−T T ) . Note thatH(ϕ) is containedϕ ϕ
2contractively in H and the inner product is defined in order to make (Id−
1/2 2T T ) a partial isometry of H ontoH(ϕ). The norm ofH(ϕ) will be denotedϕ ϕ
byk·k .ϕ
2For λ∈ , we let k denote the kernel function for the functionnal on H ofλ
−1evaluation at λ; it is given by k (z) = (1−λz) (z ∈ ) and satisfies f(λ) =λ
2 2hf,k i (f ∈ H ). Since H(ϕ) is contained contractively in H , the restrictionλ 2
to H(ϕ) of evaluation at λ is a bounded linear functionnal on H(ϕ). It is thus
ϕinduced, relative to the inner product inH(ϕ), by a vector k inH(ϕ). It is easy
λ
ϕ
to see ([19], (II-3)) that k = (Id−T T )k andϕ ϕ λλ
ϕ
f(λ)=hf,k i ,ϕλ
for all f ∈H(ϕ). From now on, b will be a nonconstant function in the unit ball
∞of H , that is an holomorphic and bounded function in , withkbk 6 1. Then∞
since T k =b(λ)k , we haveλ λb
1−b(λ)bbk =(Id−T T )k = .b λλ b
1−λz
2It is easy to see that H(b) is a closed subspace of H if and only if T is ab
partial isometry. That happens if and only if b is an inner function, that is a
2




∞function in H whose radial limits are of modulus one almost everywhere. Then
2H(b) is the orthogonal complement of the Beurling invariant subspace bH , the
typical nontrivial invariant subspace of the shift operator S. Hence, the space
H(b), with b inner, are the nontrivial invariant subspaces of the backward shift
∗S . In this case, starting with the work of S.V. Hruscev, N.K. Nikolski and
B.S. Pavlov, a whole direction of research has investigated geometric properties
of reproducting kernels inH(b) (see [4], [9], [10], [11]). One of the motivation to
study geometric properties of reproducting kernels inH(b) is the link being with
nontrigonometric exponentials systems. Recall that in the special case where
z+1 bb(z) = exp(a ), a > 0, the reproducing kernels k , with λ ∈ , arise as
λz−1
1+λthe range of the exponential functions exp(−iμw)χ , with μ = i , under a(0,a) 1−λ
2natural unitary map going fromL (0,a) toH(b). Geometric properties of family
of exponentials arise in many problems such as scaterring theory, controllability
and analysis of convolution equations (see [3] and [11] for details). We intend
to provide a comprehensive treatment of geometric properties of reproducing
kernels of H(b), emphasing the parallel with the particular case where b is an
inner function.
We now recall some basic definitions concerning geometric properties of se-
quences in an Hilbert space. For most of the definitions and facts below, one can
use [14] as a main reference.
Let H be a complex Hilbert space. If (x ) ⊂H, we denote by Span(x :n n>1 n
n> 1) the closure of the linear hull generated by (x ) . The sequence (x )n n>1 n n>1
is called:
(Co) complete if Span(x :n> 1)=H;n
(M) minimal if for all n> 1, x 6∈ Span(x :m =n);n m

xn
(UM) uniformly minimal if inf dist ,Span(x :m =n) >0;m
n>1 kx kn
(UBS) an unconditionnal basis in its closed linear span if every element x ∈
Span(x : n > 1) can be uniquely decomposed in an uncondionnal con-n X
vergent series x = a x ;n n
n>1
(RS) a Riesz basis in its closed linear span if there are positive constants c,C
such that X X X
22 2 c |a | 6 a x 6 C |a | , (1)n n n n
n>1 n>1 n>1
finite complex sequences (a ) ;n n≥1
(UB) an unconditionnal basis of H if it is complete and an unconditionnal basis
in its closed linear span.
3
66Obviously we have
(UB) =⇒ (RS)=⇒ (USB) =⇒ (UM) =⇒ (M).
In general, all the converse implications are false but K¨othe-Topelitz theorem
asserts that ifkx k 1, then (USB)⇐⇒ (RS).n
The Gram matrix of the sequence (x ) is Γ = (hx ,x i) . Uncondi-n n≥1 n m n,m≥1
tionnal basis are characterized by the fact that Γ defines an invertible operator
2on ` .
2We recall some well-known facts concerning reproducing kernels in H . Let
kλnΛ = (λ ) be a sequence of distinct points in and denote by x =n n>1 n
kk kλ 2n
the normalized reproducing kernel. Then we have
– (k ) isminimalifandonlyif(λ ) isBlaschkesequence(whichmeansλ n>1 n n>1n P Q
that (1−|λ |) < ∞). As usual, we denote by B = B = b ,n Λ λnn>1 n>1
|λ |n λ −znwhere b (z) = .λn λn 1−λ zn
– If (λ ) is a Blaschke sequence, then (k ) is complete inH(B).n n>1 λ n>1n
– (x ) is a Riesz basis ofH(B)if andonly if itis uniformly minimal whichn n>1
is equivalent to (λ ) satisfies the Carleson conditionn n>1
inf|B (λ )|>0,n n
n>1
where B =B/b ; we will write in this case (λ ) ∈ (C).n λ n n>1n
In this paper, we intend to study the property of unconditionnal basis for se-
quences of reproducing kernels inH(b). The study of the spacesH(b) frequently
∞bifurcates into two cases depending b is an extreme point of the unit ball of H
or not. We will show that for the property of unconditionnal basis inH(b), there
exists a dichotomy between the two cases. Recall that K. de Leeuw and W.
∞Rudin [7] proved that b is an extreme point of the unit ball of H (abbreviated
by b is extreme) if and only if
Z
2log(1−|b| )dm=−∞.
We now precise some notations that will be used in this paper. For a positive
2finite Borel measure ν on and q a function in L (ν), we let
Z
iθq(e ) iθ(K q)(z) := dν(e ), z∈ \ ,ν −iθ1−e z
2and we think ofK as a linear transformation ofL (ν) into the space of holomor-ν
2 nphic functions in \ . Moreover, we let H (ν) be the closed linear span of z ,
4






2n> 0, (for the norm of L (ν)) and we denote by Z the operator of multiplica-ν
2tion by the independant variable on H (ν). If ν is absolutely continuous and ρ is