A Proof of Nehari’s Theorem through
the Coefficient of a Matricial Measure
†Una Prueba del Teorema de Nehari a trav´
es
del Coeficiente de una Medida Matricial
Marisela Dom´ınguez
([email protected])Dpto. de Matem´aticas. Facultad de Ciencias. Universidad Central de Venezuela.
Apartado Postal 47159. Caracas 1041-A, Venezuela
Abstract
An introduction to generalized interpolation problems is given. Nehari’s theorem and Sarason’s commutation theorem are ob-tained. The proofs are simple and they are obtained using a generalization of the cosine of an angle.
Key words and phrases: interpolation, commutation, Hankel operator, coefficient of a matricial measure.
Resumen
Se da una introducci´on a problemas de interpolaci´on genera-lizada. Se obtienen el teorema de Nehari y el teorema de conmu-taci´on de Sarason. Las pruebas son simples y se obtienen usando una generalizaci´on del coseno del ´angulo.
Palabras y frases clave: interpolaci´on, conmutaci´on, operador de Hankel, coeficiente de una medida matricial.
†Recibido 98/05/09. Revisado 98/12/07. Aceptado 98/12/15.
1
Introduction
Several classical questions concerning moment problems, Toeplitz and Han-kel operators, weighted inequalities for the Hilbert transform and prediction theory are closely related. And the theory of analytic functions in the circle gives the natural environment for these problems.
In this paper the coefficient of a matricial measure is introduced, as a generalization of the cosine of an angle. This coefficient is used to give a char-acterization of the bounded Hankel operators, and to compute the distance of a function f in L∞ to H∞. As a consequence a representation theorem of operators commuting with special contractions is obtained.
Much function theory in the circleT≈[0,2π] is considered to depend on group properties of the circle and its dual Z. The results given in this note can be extended: Tcan be replaced by the bidimensional torus and the usual order inZcan be replaced by the lexicographic order (although some changes must be done). Details will be given in [4].
2
The coefficient of a matricial measure
Let dx be the Lebesgue measure in T. For 1 ≤ p ≤ ∞ let Lp(T) be the Lebesgue space and let k.kp be the norm inLp(T).
For n ∈ Z set en(x) = einx and for f ∈ L1(T) consider the Fourier coefficients: fb(n). The trigonometric polynomials are functions f : T→C, such that suppfbis finite and
f(x) =X n∈Z
b
f(n)en(x).
LetP be the space of trigonometric polynomials. Consider the following sets:
P1= n
f ∈ P: suppfb⊆ {n∈Z:n≥0}o,
P2= n
f ∈ P: suppfb⊆ {n∈Z:n <0}o.
If µ = (µαβ)α,β=1,2 is a 2×2 matrix of complex finite Borel measures
defined onT, then it is said thatµis acomplex finite Borel matricial measure
onT.
Letµbe a matricial measure on T. We shall say thatµ∈ D ifµis her-mitian, µ11=µ22 andµ11 is absolutely continuous with respect to Lebesgue measure.
Definition 1. The coefficient of a matricial measureµ∈ Dis
ρ(µ) = sup| Z
T
f1f2dµ12|,
where the supremum is taken over all f1 ∈ P1, f2∈ P2 such that kf1kw11 =
kf2kw11 = 1 and dµ11(x) = dµ22(x) = w11(x)dx (k.kw11 is the norm in
L2(T, w 11dx)).
Observe that ifµ11=µ12thenρ(µ) is the cosine of the angle betweenP1 andP2 inL2(T, µ11dx).
LetC(T) be the Banach space of complex continuous functions onTwith thek.k∞ norm.
Letµ∈ Dand consider the following formBµ:
Bµ(f1, f2) = 2 X
α,β=1 Z
T
fαfβdµαβ,
forf1,f2∈C(T).
LetHpbe the set of functionsf ∈Lp(T) such thatfb(n) = 0 for alln <0 (1≤p≤ ∞). These are the usual Hardy spacesHp.
An easy calculation establishes the following result:
Proposition 1. Letµandνbe two matricial measures onT. ThenBµ(f1, f2) =Bν(f1, f2)for every (f1, f2)∈ P1× P2 if and only if there ish∈H1 such
that ν11=µ11,ν22=µ22 andν12=µ12+hdx.
Ifr∈(0,1], h∈H1 andµ is a matricial measure onT, defineµ(r, h) as the matricial measure given by µ(r, h)11=rµ11,µ(r, h)22=rµ22,µ(r, h)12=
µ12+hdxandµ(r, h)21=µ21+hdx.
Proposition 2. Let r∈(0,1]andµ∈ D. Then:
Bµ(r,0)(f1, f2)≥0 for every(f1, f2)∈ P1× P2 if and only ifρ(µ)≤r. The proof follows from a simple calculation.
Proposition 3. Let µbe a matricial measure on T. Then the following con-ditions are equivalent:
(a) µis a positive matricial measure. (b) Bµ(f1, f2)≥0 for every f1,f2∈C(T).
(c) Bµ(f1, f2)≥0 for every f1,f2∈ P.
Proof. We first show that (a) implies (b). Letµbe a positive matricial mea-sure. Givenf1,f2∈C(T), there are two sequences of simple functions{g1n}n and{g1n}n such that forα= 1,2:
fα(t) =limn→∞gαn(t).
For each n let{Ani}i and {Bnk}k be the sets which occur in the canonical representations of{g1n}nand{g2n}n. LetAn0andBn0be the sets whereg1n andg2nare zero. Then the sets{∆nj}jobtained by taking all the intersections
Ani∩Bnk form a finite disjoint collection of measurable sets, and forα= 1,2
h1 and h2 can be approximated by continuous functions and these can be approximated by trigonometric polynomials. Therefore
2 X
α,β=1
Theorem 1. Let r ∈ (0,1] and µ ∈ D. Then the following conditions are equivalent:
(a) ρ(µ)≤r.
(b)µ12is absolutely continuous with respect to the Lebesgue measure and there
existsh∈H1 such that
|w12(x) +h(x)|≤rw11(x) a.e.,
where dµαβ(x) =wαβ(x)dx, for α, β= 1,2.
Proof. Ifρ(µ)≤r, thenBµ(r,0)(f1, f2)≥0 for every (f1, f2)∈ P1× P2. From the lifting property (see [3]), there existsh∈H1such thatBµ(r,h)(f1, f2)≥0 for every f1, f2∈ P. Thenµ(r, h) is positive. Therefore
|µ12(∆) + Z
∆
h(x)dx|≤rµ11(∆).
If|∆|= 0 thenµ11(∆) = 0, thusµ12(∆) = 0. So if we setdµαβ(x) =wαβ(x)dx forα, β= 1,2 then
|w12(x) +h(x)|≤rw11(x) a.e.
3
The theorem of Nehari
The shift is the operator S in L2(T) given by (Sf)(x) = eixf(x) for all
f ∈L2(T).
Let H2− be the set of functions f ∈ L2(T) such that fb(n) = 0 for all
n≥0 and letP− be the orthogonal projection inL2(T) with rangeH2−.
Definition 3. A linear operatorΓ :H2→H2− such thatP−SΓ = ΓS |H2 is
called a Hankel operator.
Proposition 4. Let Γ :H2→H2− be a bounded linear operator. Then the
following conditions are equivalent:
(a) Γ is a Hankel operator.
(b) There is a sequence {An}n∈Z such that hΓek, e−ji = Ak+j for every
Proof. Suppose that (a) holds. Givenk≥0,j >0, considern=j−1. Then
n≥0 and
hΓek, e−ji = hΓek, e−n−1i=hΓek, S−ne−1i=hSnΓek, e−1i = hP−SnΓek, e−1i=hΓSnek, e−1i=hΓek+n, e−1i = hΓek+j−1, e−1i.
Define An=hΓen−1, e−1i.
Suppose on the other hand that (b) holds. Fork≥0 andj >0 we have hP−SΓek, e−ji = hSΓek, e−ji=hΓek, S−1e−ji=hΓek, e−j−1i
= Ak+j+1=hΓek+1, e−ji=hΓSek, e−ji. ThusP−SΓ = ΓS|
H2.
Ifγ ∈L2(T) let Mγ be the multiplication operator given by Mγf =γf for allf ∈L2(T).
Proposition 5. Let Γ :H2→H2− be a bounded Hankel operator.
Then there exists γ0∈L2(T)such that Γ =P−Mγ0. Proof. Letγ0= Γe0. Thenγ0∈L2(T) and
P−Mγ0en=P
−Snγ
0=P−SnΓe0= ΓSne0= Γen. Therefore Γ =P−M
γ0.
Proposition 6. Forγ∈L2(T)letΓ :H2→H2− be defined byΓ =P−M γ
Then Γ is a Hankel operator. Even more hΓek, e−ji=bγ(−k−j), for every
k≥0andj >0. Proof. Letk≥0,j >0
hP−SΓek, e−ji = hSΓek, e−ji=hΓek, S−1e−ji=hΓek, e−j−1i = hP−γek, e−j−1i=hγek, e−j−1i=he1γek, e−ji = hMγek+1, e−ji=hP−Mγek+1, e−ji=hΓek+1, e−ji = hΓSek, e−ji
ThusP−SΓ = ΓS|H2. And
Remark. For γ ∈ L∞(T) we can consider Γ : H2 → H2− as the operator defined by Γ =P−M
γ. Then
kΓk=kP−Mγk ≤ kMγk=kγk∞.
Nehari’s theorem (see [5], [1]) says that:
Theorem 2. Let Γ :H2→H2− be a Hankel operator. IfΓ is bounded then
there exists γ∈L∞(T)such that
(a) bγ(−k−j) =hΓek, e−jifor everyk≥0,j >0
(b) Γ =P−M γ
(c) kΓk=inf{kγ−ξk∞:ξ∈H∞}=kγk∞.
Proof. (a) If Γ = 0 the result is obvious. By homogeneity we may assume that kΓk= 1.
Letγ0= Γe0 then Γ =P−Mγ0.
Consider the matricial measuresµ11=µ22=dxandµ12=µ21=−γ0dx. Thenρ(µ) =kΓk.
From Theorem 1 it follows that there existsh∈H1 such that
|γ0(x) +h(x)|≤1 a.e.
Defineγ=γ0+h. Thusγ∈L∞(T) and thenbγ(−n) =γb0(−n) for every
n≥0. Therefore for every k≥0,j >0:
b
γ(−k−j) =γb0(−k−j) =hΓek, e−ji.
It is clear that Γ =P−M γ0 =P
−M
γ0+h=P
−M γ and
inf{kγ−ξk∞:ξ∈H∞} ≤ kγk∞=kγ0+hk∞≤1 =kΓk.
On the other hand ifξ∈H∞then
kΓk=kP−Mγk=kP−Mγε−ξk ≤ kMγ−ξk=kγ−ξk∞.
Therefore
4
Generalized interpolation in
H
∞For every closed subspace K of H2 let PK be the orthogonal projection in
L2(T) with rangeK.
Definition 4. ψis said to be aninner functionifψ∈H∞ and|ψ|= 1a. e. Let⊖stand for orthogonal difference.
Proposition 7. Let ψ be a non constant inner function and let K be the closed manifold ofH2given byK=H2⊖ψH2. Givenφ∈H∞we define the operator X =PKMφ. Then X commutes with PKS|K.
The converse of the last proposition is given by the celebrated interpola-tion theorem of Sarason (see [6], [1], [2]), which says that:
Theorem 3. Given a nonconstant inner function ψletK be the closed sub-space of H2 given byK =H2⊖ψH2. If X is a bounded linear operator on
K such thatX commutes withPKS|K then there is a functionφ∈H∞ such
that X =PKMφ andkXk=kφk∞ .
Proof. Letf ∈L2(T). First we prove thatP
Kf =ψP−ψf. In order to do that, let h∈ K. Since K =H2⊖ψH2 ⊂(ψH2)⊥ =ψH2−, we have that
h∈ψH2−. Thusψh∈H2−. Using this we obtain that hh, ψP−ψfi=hψh, P−ψfi=hψh, ψfi=hh, fi.
So PKf = ψP−ψf. Let Γ : H2 → H2− be the operator defined by Γ =
ψXPK. Then
kΓk=kXPKk=kXk.
Let f ∈H2; since ψPKf =P−ψf and X commutes withPKS |K it follows that
ΓSf =ψXPKSf =ψPKSXPKf =P−ψSXPKf =P−SψXPKf =P−SΓf. Thus P−SΓ = ΓS |
H2. From Nehari’s theorem it follows that there exists
γ∈L∞(T) such that:
(a) bγ(−k−j) =hΓek, e−jifor every k≥0, j >0 (b) Γ =P−Mγ
(c) kΓk=inf{kγ−ξk∞:ξ∈H∞}=kγk∞. Since
we have thatγψ∈H∞. Setφ=γψ. FromkXk=kΓkandkγk
∞=kγψk∞= kφk∞ it is clear thatkXk=kφk∞. Letg∈K. Then
Xg = XPKg=ψΓg=ψP−Mγg=ψP−γg=ψP−ψψγεg = PKψγg=PKφg=PKMφg.
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