Spectral inverse problems for compact Hankel operators

0.28 MB·English

by  Patrick Gerard

SPECTRAL INVERSE PROBLEMS FOR COMPACT
HANKEL OPERATORS
PATRICK GE´RARDAND SANDRINEGRELLIER
Abstract. Given two arbitrary sequences (λ ) and (µ ) of real
j j≥1 j j≥1
2 numberssatisfying
1 |λ |>|µ |>|λ |>|µ |>···>|λ |>|µ |→0 ,
1 1 2 2 j j
0
2 weprovethat thereexists auniquesequencec=(cn)n∈Z+, real valued,
such that the Hankel operators Γc and Γc˜ of symbols c = (cn)n≥0 and
n c˜=(c ) respectively,areselfadjoint compactoperatorsonℓ2(Z )
a n+1 n≥0 +
and have the sequences (λ ) and (µ ) respectively as non zero
J j j≥1 j j≥1
eigenvalues. Moreover,wegiveanexplicitformulaforcandwedescribe
4
thekernelofΓ andofΓ intermsofthesequences(λ ) and(µ ) .
2 c c˜ j j≥1 j j≥1
More generally, given two arbitrary sequences (ρ ) and (σ ) of
j j≥1 j j≥1
positive numberssatisfying
]
P
ρ >σ >ρ >σ >···>ρ >σ →0 ,
A 1 1 2 2 j j
. wedescribetheset ofsequencesc=(cn)n∈Z+ ofcomplex numberssuch
h that the Hankel operators Γ and Γ are compact on ℓ2(Z ) and have
c c˜ +
t sequences(ρ ) and (σ ) respectively as non zero singular values.
a j j≥1 j j≥1
m
[

1 1. Introduction
v
1 Let c= (c ) be a sequence of complex numbers. The Hankel operator
n n≥0
7 Γ of symbol c is formally defined on ℓ2(Z ) by
9 c +
4 ∞
. x =(x ) ℓ2(Z ) , Γ (x) = c x .
1 ∀ n n≥0 ∈ + c n n+p p
0 p=0
X
2
These operators frequently appear in operator theory and in harmonic anal-
1
ysis, and we refer to the books by Nikolskii [10] and Peller [13] for an intro-
:
v
duction and their basic properties. By a well known theorem of Nehari [9],
i
X Γ is well defined and bounded on ℓ2(Z ) if and only if there exists a func-
c +
r tion f L∞(T) such that n 0,fˆ(n) = c , or equivalently if the Fourier
a series u∈ = c einx be∀lon≥gs to the spance BMO(T) of bounded mean
c n≥0 n
oscillation functions. Moreover, by a well known result of Hartman [4], Γ
c
is compact ifPand only if there exists a continuous function f on T such that
n 0,fˆ(n) = c , or equivalently if u belongs to the space VMO(T) of
n c
∀ ≥
vanishing mean oscillation functions. Assume moreover that the sequence c
is real valued. Then Γ is selfadjoint and compact, so it admits a sequence
c
of non zero eigenvalues (λ ) , tending to zero. A natural inverse spectral
j j≥1
problem is the following: given any sequence (λ ) , tending to zero, does
j j≥1
Date: January 24, 2012.
2010 Mathematics Subject Classification. 47B35, 37K15.
The second author acknowledges the support of the ANR project AHPI (ANR-07-
BLAN-0247-01).
1
SPECTRAL INVERSE PROBLEMS 2
there exist a compact selfadjoint Hankel operator Γ having this sequence
c
as non zero eigenvalues, repeated according to their multiplicity?
A complete answer to this question can be found in the literature as a
consequence of a more general theorem by Megretskii, Peller, Treil [8] char-
acterizing selfadjoint operators which are unitarily equivalent to bounded
Hankel operators. Here we state the part of their result which concerns the
compact operators.
Theorem 1 (Megretskii, Peller, Treil [8]). Let Γ be a compact, selfadjoint
operator on a separable Hilbert space. Then Γ is unitarily equivalent to a
Hankel operator if and only if the following conditions are satisfied
(1) Either ker(Γ) = 0 or dimker(Γ) = ;
(2) For any λ R∗,{d}imker(Γ λI) d∞imker(Γ+λI) 1.
∈ | − − | ≤
As a consequence of this theorem, any sequence of real numbers with
distinct absolute values and converging to 0 is the sequence of the non zero
eigenvalues of some compact selfadjoint Hankel operator.
In this paper, we are interested in finding additional constraints on the
operator Γ which give rise to uniqueness of c. With this aim in view, we
c
introduce the shifted Hankel operator Γ , where c˜ := c for all n Z .
c˜ n n+1 +
∈
If we denote by (λ ) the sequence of non zero eigenvalues of Γ and by
j j≥1 c
(µ ) thesequenceofnonzeroeigenvaluesofΓ ,onecancheck–seebelow–
j j≥1 c˜
that
λ µ λ µ 0 .
1 1 2 2
| | ≥ | |≥ | | ≥ | | ≥ ··· ≥ ··· →
Our result reads as follows.
Theorem 2. Let (λ ) , (µ ) be two sequences of real numbers tending
j j≥1 j j≥1
to zero so that
λ > µ > λ > µ > ... > 0 .
1 1 2 2
| | | | | | | | ··· →
There exists a unique real valued sequence c = (c ) such that Γ and Γ are
n c c˜
compact selfadjoint operators, the sequence of non zero eigenvalues of Γ is
c
(λ ) , and the sequence of non zero eigenvalues of Γ is (µ ) .
j j≥1 c˜ j j≥1
Furthermore, the kernel of Γ is reduced to zero if and only if the following
c
conditions hold,
∞ µ2 1 N µ2
j j
(1) 1 = , sup = .
− λ2 ∞ λ2 λ2 ∞
j=1 j! N N+1 j=1 j
X Y
Moreover, in that case, the kernel of Γ is also reduced to 0.
c˜
In complement to the above statement, let us mention that an explicit
formula for c is available, as well as an explicit description of the kernel of
Γ when it is non trivial — see Theorems 3 and 4 below.
c
Theorem 2 is in fact a consequence of a more general result concerning
the singular values of non necessarily selfadjoint compact Hankel operators.
Recall that the singular values of a bounded operator T on a Hilbert space
, are given by the following min-max formula. For every m 1, denote
H ≥
by the set of linear subspaces of of dimension at most m. The m-th
m
F H
singular value of T is given by
SPECTRAL INVERSE PROBLEMS 3
(2) s (T) = min max T(f) .
m
F∈Fm−1f∈F⊥,kfk=1k k
In this paper, we construct a homeomorphism between some set of sym-
bols c and the singular values of Γ and Γ up to the choice of an element
c c˜
in an infinite dimensional torus.
In order to state this general result we complexify and reformulate the
problem in the Hardy space. We identify ℓ2(Z ) with
+
∞ ∞
L2(T)= u : u= uˆ(n)einx , uˆ(n)2 < +
+ { | | ∞ }
n=0 n=0
X X
and we denote by Π the orthogonal projector from L2(T) onto L2(T).
+
Here and in the following, for any space of distributions E on T, the
notation E stands for the subspace of E consisting of those elements u
+
of E such that uˆ(n) = 0 for every n < 0, or equivalently which can be
holomorphically extended to the unit disc. In that case, we will still denote
by u(z) the value of this holomorphic extension at the point z of the unit
disc.
We endow L2(T) with the scalar product
+
dx
(uv) := uv
| 2π
ZT
and with the associated symplectic form
ω(u,v) = Im(uv) .
|
For u sufficiently smooth, we define a C-antilinear operator on L2 by
+
H (h) = Π(uh) , h L2 .
u ∈ +
If u= u ,
c
H\(h)(n)= Γ (x) , x := hˆ(p) .
u c n p
Because of this equality, H is called the Hankel operator of symbol u.
u
Similarly, Γ corresponds to the operator K = H T where T denotes
c˜ u u z z
multiplicationbyz. RemarkthatbydefinitionH = H . Inthefollowing,
u Π(u)
we always consider holomorphic symbols u = Π(u).
As stated before, by the Nehari theorem ([9]), H is well defined and
u
bounded on L2(T) if and only if u belongs to Π(L∞(T)) or to BMO (T).
+ +
Moreover, bytheHartmantheorem([4]),itisacompactoperatorifandonly
if u is the projection of a continuous function on the torus, or equivalently
if and only if it belongs to VMO (T) with equivalent norms. Furthermore,
+
remark that this operator H is selfadjoint as an antilinear operator in the
u
sense that for any h ,h L2,
1 2 ∈ +
(h H (h )) = (h H (h )).
1 u 2 2 u 1
| |
A crucial property of Hankel operators is that H T = T∗H so that, in
u z z u
particular,
(3) K2 =H2 ( u)u.
u u − ·|
SPECTRAL INVERSE PROBLEMS 4
Assume u VMO (T) and denote by (ρ ) the sequence of singular
+ j j≥1
∈
values of H labelled according to the min-max formula (2). Since, via the
u
Fourier transform, H2 identifies to Γ Γ∗ with c = uˆ, (ρ ) is also the
u c c j j≥1
sequence of singular values of Γ . Similarly, K is a compact, so it has
uˆ u
a sequence (σ ) of singular values tending to 0, which are the singular
j j≥1
valuesofΓ ,sinceK2 identifiestoΓ Γ∗. FromEquality(3)andthemin-max
c˜ u c˜ c˜
formula (2), one obtains
ρ σ ρ σ 0.
1 1 2 2
≥ ≥ ≥ ≥ ··· ≥ ··· →
We denote by VMO the set of u VMO (T) such that H and K
+,gen + u u
∈
admit only simple singular values with strict inequalities, or equivalently
such that H2 and K2 := H2 ( u)u admit only simple positive eigenvalues
u u u− ·|
ρ2 > ρ2 > > 0 and σ2 > σ2 > > 0 so that
1 2 ··· ··· → 1 2 ··· ··· →
ρ2 > σ2 > ρ2 > σ2 > > 0.
1 1 2 2 ··· ··· →
For any integer N, we denote by (2N) the set of symbol u such that
V
the rank of H and the rank of K are both equal to N. By a theorem of
u u
Kronecker(see[5]), (2N)isacomplexmanifoldofdimension2N consisting
V
of rational functions. One can consider as well the set (2N 1) of symbols
V −
u such that H is of rank N and K is of rank N 1. It defines a complex
u u
−
manifold of rational functions of complex dimension 2N 1.
−
By the arguments developed in [2], it is straightforward to verify that
VMO is a dense G subset of VMO (T). Indeed, let us consider the
+,gen δ +
set which consists of functions u VMO (T) such that the N first
N +
U ∈
eigenvalues of H2 and of K2 are simple. This set is obviously open in
u u
VMO (T). Moreover, in Lemma 4 of [2], it is proved that (2N) :=
+ N
U ∩V
(2N) is a dense open subset of (2N). Now any element u in VMO
gen +
V V
may beapproximated by an element in (2N′), N′ > N, which can be itself
V
approximated by an element in (2N′) , since N′ N. Eventually,
gen N
V ⊂ U ≥
VMO is the intersection of the ’s which are open and dense, hence
+,gen N
U
VMO is a dense G set.
+,gen δ
Let u VMO . Denote by ((ρ ) the singular values of H and by
+,gen j j≥1 u
∈
(σ ) the singular values of K . Using the antilinearity of H there exists
j j≥1 u u
an orthonormal family (e ) of the range of H such that
j j≥1 u
H (e ) = ρ e , j 1.
u j j j
≥
Notice that the orthonormal family is determined by u up to a change of
sign on some of the e . We claim that (1e ) = 0. Indeed, if (1e ) = 0 then
j j j
| 6 |
(ue ) = ρ (e 1) = 0 and, in view of (3), ρ2 would be an eigenvalue of K2,
| j j j| j u
which contradicts the assumption. Therefore we can define the angles
ϕ (u) := arg(1e )2 j 1 .
j j
| ≥
We do the same analysis with the operator K = H T . As before, by the
u u z
antilinearity of K there exists an orthonormal family (f ) of the range
u j j≥1
of K such that
u
K (f )= σ f , j 1,
u j j j
≥
and the family is determined by u up to a change of sign on some of the f .
j
One has also (uf ) = 0 because of the assumption on the ρ ’s and σ ’s. We
j j j
| 6
SPECTRAL INVERSE PROBLEMS 5
set
θ (u) := arg(uf )2, j 1 .
j j
| ≥
Our main result is the following.
Theorem 3. The mapping
χ := u VMO ζ = ((ζ = ρ e−iϕj) ,(ζ = σ e−iθj) )
+,gen 2j−1 j j≥1 2j j j≥1
∈ 7→
is a homeomorphism onto
Ξ := (ζ ) CZ+, ζ > ζ > ζ > ζ > 0 .
j j≥1 1 2 3 4
{ ∈ | | | | | | | |··· ··· → }
Moreover, one has an explicit formula for the inverse mapping. Namely, if
ζ is given in Ξ, then the Fourier coefficients of u are given by
(4) uˆ(n)= X.AnY ,
where A= (A ) is the bounded operator on ℓ2 defined by
jk j,k≥1
∞ ν ν ζ κ2 ζ
(5) A = j k 2k−1 m 2m , j,k 1 ,
jk (ζ 2 ζ 2)(ζ 2 ζ 2) ≥
2j−1 2m 2k−1 2m
m=1 | | −| | | | −| |
X
with
σ2 ρ2 σ2
(6) ν2 := 1 j j − k ,
j − ρ2 ρ2 ρ2
j !k6=j j − k!
Y
σ2 ρ2
(7) κ2 := ρ2 σ2 m− ℓ ,
m m− m σ2 σ2
ℓ6=m(cid:18) m− ℓ(cid:19)
(cid:0) (cid:1) Y
(8) X =(ν ζ ) , Y = (ν ) ,
j 2j−1 j≥1 j j≥1
and
∞
V.W := v w if V = (v ) ,W = (w ) .
j j j j≥1 j j≥1
j=1
X
Theorem 3 calls for several comments. Firstly, it is not difficult to see
that the first part of Theorem 2 is a direct consequence of Theorem 3 (see
the end of Section 3 below). More generally, as an immediate corollary of
Theorem 3, one shows that, for any given sequences (ρ ) and (σ )
j j≥1 j j≥1
satisfying
ρ > σ > ρ > σ > 0,
1 1 2 2
··· →
thereexistsaninfinitedimensionaltorusofsymbolscsuchthatthe(ρ ) ’s
j j≥1
are the non zero singular values of Γ , and the (σ ) ’s are the non zero
c j j≥1
singular values of Γ .
c˜
Next we make the connection with previous results. In a preceding arti-
cle ([3]), we have obtained an analogue of Theorem 3 in the more restricted
context of Hilbert-Schmidt Hankel operators. This result arises in [3] as
a byproduct of the study of the dynamics of some completely integrable
Hamiltonian system called the cubic Szego¨ equation (see [2] and [3]). In
this setting the phase space of this Hamiltonian system is the Sobolev space
1/2
H , which is the space of symbols of Hilbert-Schmidt Hankel operators,
+
and the restriction of the mapping χ to the phase space can be interpreted
SPECTRAL INVERSE PROBLEMS 6
as an action-angle map. In the present paper, we extend this result to com-
pact Hankel operators, which is the natural setting for an inverse spectral
problem.
Finally, wewouldliketocommentabouttheaboveexplicitformulagiving
uˆ(n). The boundedness of operator A defined by (5) is not trivial. In fact,
it is a consequence of the proof of the theorem. However, it is possible
to give a direct proof of this boundedness, see Appendix 2. Furthermore,
from the complicated structure of formula (4), it seems difficult to check
directly that the corresponding Hankel operators have the right sequences
of singular values, namely that the map χ is onto. Our proof is in fact
completely different and is based on some compactness argument, while, as
in [3], the explicit formula is only used to establish the injectivity of χ.
We now state our last result, which describes the kernel of H in terms
u
of the ζ = χ(u).
AskerH isinvariantbytheshift,theBeurlingtheorem—seee.g. [14]—
u
provides the existence of an inner function ϕ so that kerH = ϕL2. We use
u +
the notation of Theorem 3 to describe ϕ. Denote by R the range of H .
u
Theorem 4. We keep the notation of Theorem 3. Let u VMO . The
+,gen
∈
kernel of H and the kernel of K are reduced to zero if and only if 1 R R
u u
∈ \
or if and only if the following conditions hold.
∞ σ2 1 N σ2
j j
(9) 1 = , sup = .
− ρ2 ∞ ρ2 ρ2 ∞
j=1 j ! N N+1 j=1 j
X Y
When these conditions are not satisfied, kerH = ϕL2 with ϕ inner satis-
u +
fying
(1) if 1 does not belong to the closure of the range of H i.e. 1 / R,
u
∈
then
ϕ(z) = (1 ν2)−1/2(1 α zn)
− j − n
n≥0
X X
where
(10) α = Y.AnY
n
Furthermore, kerK = kerH = ϕL2.
u u +
(2) if 1 belongs to the range of H , i.e. 1 R, then ϕ(z) =zψ(z) with
u
∈
−1/2
∞ ν2
ψ(z) = j β zn
 ρ2 n
j=1 j n≥0
X X
 
where
(11) β = W.AnY , W = (ν ζ ρ−2) .
n j 2j−1 j j≥1
Furthermore, kerK = kerH CH−1(1) = ϕL2 Cψ.
u u⊕ u + ⊕
We end this introduction by describing the organization of this paper.
In Section 2, we start the proof of Theorem 3. We first recall from [3]
a finite dimensional analogue to Theorem 3. Then we generalize from [3]
an important trace formula to arbitrary compact Hankel operators. We
then use this formula and the Adamyan-Arov-Krein theorem to derive a
SPECTRAL INVERSE PROBLEMS 7
crucialcompactnesslemmaaboutHankeloperators. Usingthiscompactness
lemma, we prove Theorem 3 in Section 3, and we infer the first part of
Theorem 2. Section 4 is devoted to the proof of Theorem 4, from which the
second part of Theorem 2 easily follows. Finally, for the convenience of the
reader, we have gathered in Appendix 1 the main steps of the proof of the
finite dimensional analogue of Theorem 3, while Appendix 2 is devoted to a
direct proof of the boundedness of operator A involved in Theorem 3.
2. Preliminary results
TheproofofTheorem3isbasedonafiniterankapproximationofH . We
u
firstrecallthenotation andasimilarresultobtainedonfiniterankoperators
in [3].
2.1. The finite rank result. By a theorem due to Kronecker ([5]), the
Hankel operator H is of finite rank if and only if u is a rational function,
u
holomorphic in the unit disc. As in the introduction, we consider (2N)
V
the set of rational functions u, holomorphic in the unit disc, so that H
u
and K are of finite rank N. It is elementary to check that (2N) is a
u
V
2N-dimensional complex submanifold of L2 (we refer to [2] for a complete
+
description of this set and for an elementary proof of Kronecker Theorem).
We denote by (2N) the set of functions u (2N) such that H2 and
V gen ∈ V u
K2 have simple distinct eigenvalues (ρ2) and (σ2 ) respectively
u j 1≤j≤N m 1≤m≤N
with
ρ2 > σ2 > ρ2 > ...ρ2 > σ2 > 0.
1 1 2 N N
As in the introduction, we can define new variables on (2N) and a
gen
V
corresponding mapping χ . The following result has been proven in [3].
N
Theorem 5. The mapping
χ := u (2N) ζ = (ζ = ρ e−iϕj,ζ = σ e−iθj)
N gen 2j−1 j 2j j 1≤j≤N
∈ V 7→
is a symplectic diffeomorphism onto
Ξ := ζ C2N, ζ > ζ > ζ > ζ > > ζ > ζ > 0
N 1 2 3 4 2N−1 2N
{ ∈ | | | | | | | | ··· | | | | }
in the sense that the image of the symplectic form ω by χ satisfies
N
1
(12) (χ ) ω = dζ dζ .
N ∗ j j
2i ∧
1≤j≤2N
X
There is also an explicit formula for the inverse χ analogous to the
N
one given in Theorem 3 except that the sums in formulae (4) run over the
integers 1,...,N.
In order to prove the extension of Theorem 5 to VMO , we have to
+,gen
extend some tools introduced in [3].
2.2. The functional J(x). Let H be a compact selfadjoint antilinear op-
erator on a Hilbert space . Let A = H2 and e so that e = 1.
H ∈ H k k
Notice that Ais selfadjoint, positive and compact. We definethegenerating
function of H for x small, by
| |
∞
J(x)(A) = 1+ xnJ
n
n=1
X
SPECTRAL INVERSE PROBLEMS 8
where J = J (A) = (An(e)e). Consider the operator
n n
|
B := A ( H(e))H(e)
− ·|
which is also selfadjoint, positive and compact. Denote by (a ) (resp.
j j≥1
(b ) ) the non-zero eigenvalues of A (resp. of B) labelled according to the
j j≥1
min-max principle,
a b a ...
1 1 2
≥ ≥ ≥
Notice that
J(x)(A) = ((I xA)−1(e)e)
− |
which shows that J extends as an entire meromorphic function, with poles
at x = 1 ,j 1.
aj ≥
Proposition 1.
∞
1 b x 1
j
(13) J(x)(A) = − , x / ,j 1 .
1 a x ∈ a ≥
j=1 − j (cid:26) j (cid:27)
Y
Proof. We first assume A and B in the trace class. In that case, we can
compute the trace of (I xA)−1 (I xB)−1. We first write
− − −
x
[(I xA)−1 (I xB)−1](f)= (f (I xA)−1H(e)) (I xA)−1H(e).
− − − J(x) | − · −
Consequently, taking the trace, we get
x
Tr[(I xA)−1 (I xB)−1]= (I xA)−1H(e) 2.
− − − J(x)k − k
As, on the one hand,
(I xA)−1H(e) 2 = ((I xA)−1A(e)e) = J′(x)
k − k − |
and on the other hand
Tr[(I xA)−1 (I xB)−1] = xTr[A(I xA)−1 B(I xB)−1]
− − − − − −
∞
a b
j j
= x
1 a x − 1 b x
j=1(cid:18) − j − j (cid:19)
X
we get
∞ a b J′(x) 1 1
j j
(14) = , x / , ,j 1 .
1 a x − 1 b x J(x) ∈ a b ≥
j=1(cid:18) − j − j (cid:19) (cid:26) j j (cid:27)
X
From this equation, one gets easily formula (13) for A and B in the trace
class. To extend it to compact operators, we first recall that
a b a .
j j j+1
≥ ≥
Hence, (a b )convergeswhenAiscompactsince0 a b a a
j j− j ≤ j− j ≤ j− j+1
and a tends to zero by compactness of A. Hence, the infinite product in
j
P
Formula(13)converges,andtheabovecomputationmakessenseforcompact
operators. (cid:3)
SPECTRAL INVERSE PROBLEMS 9
Lemma 1. Let e with e = 1. Let (H ) be a sequence of compact
p
∈ H k k
selfadjoint antilinear operators onaHilbertspace which convergesstrongly
H
to H, namely
h , H h Hh .
p
∀ ∈ H p−→→∞
We assume that H is compact. Let A = H2, B = A ( H (e))H (e),
p p p p − · | p p
and A= H2 et B = A ( H(e))H(e) their strong limits. For every j 1,
− ·| ≥
denote by the set of linear subspaces of of dimension at most j, set
j
F H
(p)
a = min max (A (h)h) ,
j F∈Fj−1h∈F⊥,khk=1 p |
(p)
b = min max (B (h)h) .
j F∈Fj−1h∈F⊥,khk=1 p |
Assume there exist (a ) and (b ) such that
j j
(p) (p)
sup a a 0 , sup b b 0,
j≥1| j − j| p−→→∞ j≥1| j − j|p−→→∞
and the non-zero a , b are pairwise distinct. Then the positive eigenvalues
j m
of A are simple and are exactly the a ’s; similarly, the positive eigenvalues
j
of B = A ( H(e))H(e) are simple and are exactly the b ’s.
m
− · |
Proof. By assumption, for every h , we have
∈ H
(15) A (h) A(h) .
p
p−→→∞
Since the norm of A is uniformly bounded, we conclude that (15) holds
p
uniformly for h in every compact subset of , hence
H
n 1,An(h) An(h) .
∀ ≥ p p−→→∞
In particular, for every n 1,
≥
J (A ) := (An(e)e) (An(e)e) := J (A) ,
n p p | p−→→∞ | n
and there exists C > 0 such that
n 1, supJ (A ) Cn .
n p
∀ ≥ ≤
p
Choose δ > 0 such that δC < 1. Then, for every real number x such that
x < δ, we have, by dominated convergence,
| |
∞ ∞
J(x)(A ) := 1+ xnJ (A ) 1+ xnJ (A) := J(x)(A) .
p n p n
p−→→∞
n=1 n=1
X X
On the other hand, in view of the assumption about the convergence of
(p) (p)
(a ) and (b ) and the convergence of the product in Formula (13),
j j≥1 j j≥1
we also have, for x < δ,
| |
∞ 1 b(p)x ∞ 1 b x
(16) J(x)(A ) = − j − j .
p
jY=1 1−aj(p)x! p−→→∞jY=1(cid:18)1−ajx(cid:19)
Hence, we obtain
∞
1 b x
j
(17) J(x)(A) = − .
1 a x
j=1(cid:18) − j (cid:19)
Y
SPECTRAL INVERSE PROBLEMS 10
Byassumption,thenon-zeroa ,b arepairwisedistinctsonocancellation
j m
can occur in the right hand side of (13), and the poles are all distinct.
On the other hand, denote by (a ) the family of eigenvalues of A and by
j
(b ) the one of B. By a classical result (see e.g. Lemma 1, section 2.2 of
j
[3]),
a ,j 1 a ,j 1 , b ,j 1 b ,j 1
j j j j
{ ≥ } ⊂ { ≥ } { ≥ } ⊂ { ≥ }
and the multiplicity of positive eigenvalues is 1. Consequently, there is no
cancellation in the expression of J(x)(A) and all the poles are simple. We
conclude that a = a , b = b for every j 1. (cid:3)
j j j j
≥
2.3. A compactness result. From now on, we choose = L2 and e = 1.
H +
As a first application of Proposition 1, we obtain the following.
Lemma 2. For any u VMO (T), we have
+
∈
∞ 1 xσ2(u) ∞ ρ2(u)ν2 1
J(x) := J(x)(H2)= − j = 1+x j j , x / .
u 1 xρ2(u) 1 xρ2(u) ∈ ρ2(u)
jY=1 − j Xj=1 − j ( j )j≥1
Here ν := (1e ). In particular,
j j
| | |
σ2 ρ2 σ2
ν2 = 1 j j − k
j − ρ2 ρ2 ρ2
j!k6=j j − k!
Y
The first equality is just a consequence of (13). For the second equality,
we use the formula J(x) = ((I xH2)−1(1)1) and we expand 1 according
− u |
to the decomposition
L2 = Ce kerH .
+ ⊕j≥1 j ⊕ u
From Lemma 1, we infer the following compactness result, which can be
interpreted as a compensated compactness result.
Proposition 2. Let (u ) be a sequence of VMO (T) weakly convergent to
p +
u in VMO (T). We assume that, for some sequences (ρ ) and (σ ),
+ j j
sup ρ (u ) ρ 0 , sup σ (u ) σ 0,
j≥1| j p − j|p−→→∞ j≥1| j p − j|p−→→∞
and the following simplicity assumption: allthe non-zero ρ , σ are pairwise
j m
distinct. Then, for every j 1, ρ (u) =ρ , σ (u) = σ , and the convergence
≥ j j j j
of u to u is strong in VMO (T).
p +
Remark 1. Let us emphasize that this result specifically uses the structure
of Hankel operators. It is false in general for compact operators assumed
to converge only strongly. One also has to remark that the simplicity of the
eigenvalues is a crucial hypothesis as the following example shows. Denote
by (u ), p < 1, p real, the sequence of functions defined by
p
| |
z p
u (z) = − .
p
1 pz
−
Then, the selfadjoint Hankel operators H and K have eigenvalues λ =
up up 1
µ = 1 and λ = 1 and µ = λ = 0 for m 2 independently of p . As
1 2 m m+1
− ≥
p goes to 1, p < 1, u tends weakly to the constant function 1, hence the
p
−
convergence is not strong in VMO. Indeed, H is the rank one operator
−1