Inverse pb Action-angle Proof
Inverse Spectral problem for compact
Hankel Operators
Sandrine Grellier
Université d’Orléans- Fédération Denis Poisson
SMF-VMS congress, University of Hue, August 20 - 24, 2012
Joint work with Patrick Gérard (Université Paris sud)
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Hankel operators
Given c = (cn )n≥0 a ”symbol”, define the operator Γc by
∀x = (xn )n≥0 ∈ `2 (N) , Γc (x)n =
∞
X
cn+p xp .
p=0
Let u(z) :=
P
n≥0 cn z
n.
1
Γc is of finite rank iff u is rational (Kronecker 1881).
2
Γc is Hilbert–Schmidt iff u ∈ H 1/2 (T).
3
4
5
Γc belongs to the Schatten class Sp , 0 < p < ∞ iff
1/p
u ∈ Bp,p (T) (Peller, Semmes 1984).
Γc is bounded iff ∃f ∈ L∞ (T); cn = f̂ (n), n ≥ 0 or iff
u ∈ BMO(T) (Nehari 1957).
Γc is compact iff ∃f ∈ C(T); cn = f̂ (n), n ≥ 0 or iff
u ∈ VMO(T) (Hartman 1958).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Hankel operators
Given c = (cn )n≥0 a ”symbol”, define the operator Γc by
∀x = (xn )n≥0 ∈ `2 (N) , Γc (x)n =
∞
X
cn+p xp .
p=0
Let u(z) :=
P
n≥0 cn z
n.
1
Γc is of finite rank iff u is rational (Kronecker 1881).
2
Γc is Hilbert–Schmidt iff u ∈ H 1/2 (T).
3
4
5
Γc belongs to the Schatten class Sp , 0 < p < ∞ iff
1/p
u ∈ Bp,p (T) (Peller, Semmes 1984).
Γc is bounded iff ∃f ∈ L∞ (T); cn = f̂ (n), n ≥ 0 or iff
u ∈ BMO(T) (Nehari 1957).
Γc is compact iff ∃f ∈ C(T); cn = f̂ (n), n ≥ 0 or iff
u ∈ VMO(T) (Hartman 1958).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Hankel operators
Given c = (cn )n≥0 a ”symbol”, define the operator Γc by
∀x = (xn )n≥0 ∈ `2 (N) , Γc (x)n =
∞
X
cn+p xp .
p=0
Let u(z) :=
P
n≥0 cn z
n.
1
Γc is of finite rank iff u is rational (Kronecker 1881).
2
Γc is Hilbert–Schmidt iff u ∈ H 1/2 (T).
3
4
5
Γc belongs to the Schatten class Sp , 0 < p < ∞ iff
1/p
u ∈ Bp,p (T) (Peller, Semmes 1984).
Γc is bounded iff ∃f ∈ L∞ (T); cn = f̂ (n), n ≥ 0 or iff
u ∈ BMO(T) (Nehari 1957).
Γc is compact iff ∃f ∈ C(T); cn = f̂ (n), n ≥ 0 or iff
u ∈ VMO(T) (Hartman 1958).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Hankel operators
Given c = (cn )n≥0 a ”symbol”, define the operator Γc by
∀x = (xn )n≥0 ∈ `2 (N) , Γc (x)n =
∞
X
cn+p xp .
p=0
Let u(z) :=
P
n≥0 cn z
n.
1
Γc is of finite rank iff u is rational (Kronecker 1881).
2
Γc is Hilbert–Schmidt iff u ∈ H 1/2 (T).
3
4
5
Γc belongs to the Schatten class Sp , 0 < p < ∞ iff
1/p
u ∈ Bp,p (T) (Peller, Semmes 1984).
Γc is bounded iff ∃f ∈ L∞ (T); cn = f̂ (n), n ≥ 0 or iff
u ∈ BMO(T) (Nehari 1957).
Γc is compact iff ∃f ∈ C(T); cn = f̂ (n), n ≥ 0 or iff
u ∈ VMO(T) (Hartman 1958).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Hankel operators
Given c = (cn )n≥0 a ”symbol”, define the operator Γc by
∀x = (xn )n≥0 ∈ `2 (N) , Γc (x)n =
∞
X
cn+p xp .
p=0
Let u(z) :=
P
n≥0 cn z
n.
1
Γc is of finite rank iff u is rational (Kronecker 1881).
2
Γc is Hilbert–Schmidt iff u ∈ H 1/2 (T).
3
4
5
Γc belongs to the Schatten class Sp , 0 < p < ∞ iff
1/p
u ∈ Bp,p (T) (Peller, Semmes 1984).
Γc is bounded iff ∃f ∈ L∞ (T); cn = f̂ (n), n ≥ 0 or iff
u ∈ BMO(T) (Nehari 1957).
Γc is compact iff ∃f ∈ C(T); cn = f̂ (n), n ≥ 0 or iff
u ∈ VMO(T) (Hartman 1958).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Hankel operators
Given c = (cn )n≥0 a ”symbol”, define the operator Γc by
∀x = (xn )n≥0 ∈ `2 (N) , Γc (x)n =
∞
X
cn+p xp .
p=0
Let u(z) :=
P
n≥0 cn z
n.
1
Γc is of finite rank iff u is rational (Kronecker 1881).
2
Γc is Hilbert–Schmidt iff u ∈ H 1/2 (T).
3
4
5
Γc belongs to the Schatten class Sp , 0 < p < ∞ iff
1/p
u ∈ Bp,p (T) (Peller, Semmes 1984).
Γc is bounded iff ∃f ∈ L∞ (T); cn = f̂ (n), n ≥ 0 or iff
u ∈ BMO(T) (Nehari 1957).
Γc is compact iff ∃f ∈ C(T); cn = f̂ (n), n ≥ 0 or iff
u ∈ VMO(T) (Hartman 1958).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Hankel operators
Given c = (cn )n≥0 a ”symbol”, define the operator Γc by
∀x = (xn )n≥0 ∈ `2 (N) , Γc (x)n =
∞
X
cn+p xp .
p=0
Let u(z) :=
P
n≥0 cn z
n.
1
Γc is of finite rank iff u is rational (Kronecker 1881).
2
Γc is Hilbert–Schmidt iff u ∈ H 1/2 (T).
3
4
5
Γc belongs to the Schatten class Sp , 0 < p < ∞ iff
1/p
u ∈ Bp,p (T) (Peller, Semmes 1984).
Γc is bounded iff ∃f ∈ L∞ (T); cn = f̂ (n), n ≥ 0 or iff
u ∈ BMO(T) (Nehari 1957).
Γc is compact iff ∃f ∈ C(T); cn = f̂ (n), n ≥ 0 or iff
u ∈ VMO(T) (Hartman 1958).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Inverse spectral problem for Hankel operators
Let u ∈ VMO(T) and c real valued so that Γc is compact and
selfadjoint. Non zero eigenvalues ordered so that
|ρ1 | ≥ |ρ2 | ≥ · · · ≥ |ρj | ≥ · · · → 0.
Inverse spectral problem ?
Theorem (Megretskii, Peller, Treil 1995)
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 dim ker(Γ) = ∞ ;
2
For any λ ∈ R∗ , |dim ker(Γ − λI) − dim ker(Γ + λI)| ≤ 1.
In particular assume the generic case
|ρ1 | > |ρ2 | > · · · > |ρj | > · · · → 0
∃c such that the ρj ’s are the non-zero eigenvalues of Γc .
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Inverse spectral problem for Hankel operators
Let u ∈ VMO(T) and c real valued so that Γc is compact and
selfadjoint. Non zero eigenvalues ordered so that
|ρ1 | ≥ |ρ2 | ≥ · · · ≥ |ρj | ≥ · · · → 0.
Inverse spectral problem ?
Theorem (Megretskii, Peller, Treil 1995)
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 dim ker(Γ) = ∞ ;
2
For any λ ∈ R∗ , |dim ker(Γ − λI) − dim ker(Γ + λI)| ≤ 1.
In particular assume the generic case
|ρ1 | > |ρ2 | > · · · > |ρj | > · · · → 0
∃c such that the ρj ’s are the non-zero eigenvalues of Γc .
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Inverse spectral problem for Hankel operators
Let u ∈ VMO(T) and c real valued so that Γc is compact and
selfadjoint. Non zero eigenvalues ordered so that
|ρ1 | ≥ |ρ2 | ≥ · · · ≥ |ρj | ≥ · · · → 0.
Inverse spectral problem ?
Theorem (Megretskii, Peller, Treil 1995)
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 dim ker(Γ) = ∞ ;
2
For any λ ∈ R∗ , |dim ker(Γ − λI) − dim ker(Γ + λI)| ≤ 1.
In particular assume the generic case
|ρ1 | > |ρ2 | > · · · > |ρj | > · · · → 0
∃c such that the ρj ’s are the non-zero eigenvalues of Γc .
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
The shifted Hankel operator
Let u ∈ VMO(T), we introduce
∀n ≥ 0 , c̃n := cn+1 and c̃ = (c̃n )n≥0 .
The non zero eigenvalues (σj )j≥1 of Γc̃ satisfy (min-max
principle)
|ρ1 | ≥ |σ1 | ≥ |ρ2 | ≥ |σ2 | ≥ . . .
Strict inequalities define a dense Gδ subspace of VMO(T)
denoted by VMOgen (kukVMO = kΓc k).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
A double inverse spectral theorem
Theorem (P. Gérard - S.G. 2011)
Let (ρj )j≥1 , (σj )j≥1 be sequences of real numbers satisfying
|ρ1 | > |σ1 | > |ρ2 | > |σ2 | > ... > |ρj | > |σj | · · · → 0 .
There exists a unique sequence c = (cn )n≥0 of real numbers
such that Γc and Γc̃ are compact selfadjoint operators, the
sequence of non zero eigenvalues of Γc is (ρj )j≥1 , and the
sequence of non zero eigenvalues of Γc̃ is (σj )j≥1 .
Furthermore, the kernel of Γc is reduced to zero if and only if
the following conditions hold,
!
∞
N
2
X
σj2
1 Y σj
1 − 2 = ∞,
sup 2
= ∞.
2
ρj
N ρN+1 j=1 ρj
j=1
Moreover, in that case, the kernel of Γc̃ is also reduced to 0.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
A double inverse spectral theorem
Theorem (P. Gérard - S.G. 2011)
Let (ρj )j≥1 , (σj )j≥1 be sequences of real numbers satisfying
|ρ1 | > |σ1 | > |ρ2 | > |σ2 | > ... > |ρj | > |σj | · · · → 0 .
There exists a unique sequence c = (cn )n≥0 of real numbers
such that Γc and Γc̃ are compact selfadjoint operators, the
sequence of non zero eigenvalues of Γc is (ρj )j≥1 , and the
sequence of non zero eigenvalues of Γc̃ is (σj )j≥1 .
Furthermore, the kernel of Γc is reduced to zero if and only if
the following conditions hold,
!
∞
N
2
X
σj2
1 Y σj
1 − 2 = ∞,
sup 2
= ∞.
2
ρj
N ρN+1 j=1 ρj
j=1
Moreover, in that case, the kernel of Γc̃ is also reduced to 0.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
A double inverse spectral theorem
Theorem (P. Gérard - S.G. 2011)
Let (ρj )j≥1 , (σj )j≥1 be sequences of real numbers satisfying
|ρ1 | > |σ1 | > |ρ2 | > |σ2 | > ... > |ρj | > |σj | · · · → 0 .
There exists a unique sequence c = (cn )n≥0 of real numbers
such that Γc and Γc̃ are compact selfadjoint operators, the
sequence of non zero eigenvalues of Γc is (ρj )j≥1 , and the
sequence of non zero eigenvalues of Γc̃ is (σj )j≥1 .
Furthermore, the kernel of Γc is reduced to zero if and only if
the following conditions hold,
!
∞
N
2
X
σj2
1 Y σj
1 − 2 = ∞,
sup 2
= ∞.
2
ρj
N ρN+1 j=1 ρj
j=1
Moreover, in that case, the kernel of Γc̃ is also reduced to 0.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
A double inverse spectral theorem
Theorem (P. Gérard - S.G. 2011)
Let (ρj )j≥1 , (σj )j≥1 be sequences of real numbers satisfying
|ρ1 | > |σ1 | > |ρ2 | > |σ2 | > ... > |ρj | > |σj | · · · → 0 .
There exists a unique sequence c = (cn )n≥0 of real numbers
such that Γc and Γc̃ are compact selfadjoint operators, the
sequence of non zero eigenvalues of Γc is (ρj )j≥1 , and the
sequence of non zero eigenvalues of Γc̃ is (σj )j≥1 .
Furthermore, the kernel of Γc is reduced to zero if and only if
the following conditions hold,
!
∞
N
2
X
σj2
1 Y σj
1 − 2 = ∞,
sup 2
= ∞.
2
ρj
N ρN+1 j=1 ρj
j=1
Moreover, in that case, the kernel of Γc̃ is also reduced to 0.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
A double inverse spectral theorem
Theorem (P. Gérard - S.G. 2011)
Let (ρj )j≥1 , (σj )j≥1 be sequences of real numbers satisfying
|ρ1 | > |σ1 | > |ρ2 | > |σ2 | > ... > |ρj | > |σj | · · · → 0 .
There exists a unique sequence c = (cn )n≥0 of real numbers
such that Γc and Γc̃ are compact selfadjoint operators, the
sequence of non zero eigenvalues of Γc is (ρj )j≥1 , and the
sequence of non zero eigenvalues of Γc̃ is (σj )j≥1 .
Furthermore, the kernel of Γc is reduced to zero if and only if
the following conditions hold,
!
∞
N
2
X
σj2
1 Y σj
1 − 2 = ∞,
sup 2
= ∞.
2
ρj
N ρN+1 j=1 ρj
j=1
Moreover, in that case, the kernel of Γc̃ is also reduced to 0.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
A double inverse spectral theorem
Theorem (P. Gérard - S.G. 2011)
Let (ρj )j≥1 , (σj )j≥1 be sequences of real numbers satisfying
|ρ1 | > |σ1 | > |ρ2 | > |σ2 | > ... > |ρj | > |σj | · · · → 0 .
There exists a unique sequence c = (cn )n≥0 of real numbers
such that Γc and Γc̃ are compact selfadjoint operators, the
sequence of non zero eigenvalues of Γc is (ρj )j≥1 , and the
sequence of non zero eigenvalues of Γc̃ is (σj )j≥1 .
Furthermore, the kernel of Γc is reduced to zero if and only if
the following conditions hold,
!
∞
N
2
X
σj2
1 Y σj
1 − 2 = ∞,
sup 2
= ∞.
2
ρj
N ρN+1 j=1 ρj
j=1
Moreover, in that case, the kernel of Γc̃ is also reduced to 0.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Explicit formula
Explicit formula for u =
P
n cn z
n
in VMOgen (T),
Explicit description of the kernel of Γc when it is non trivial !
Extension of the result to non necessarily self adjoint
compact Hankel operators.
Method : construct a homeomorphism between VMOgen (T) and
the singular values of Γc and Γc̃ up to a choice of an element in
an infinite dimensional torus.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Explicit formula
Explicit formula for u =
P
n cn z
n
in VMOgen (T),
Explicit description of the kernel of Γc when it is non trivial !
Extension of the result to non necessarily self adjoint
compact Hankel operators.
Method : construct a homeomorphism between VMOgen (T) and
the singular values of Γc and Γc̃ up to a choice of an element in
an infinite dimensional torus.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Explicit formula
Explicit formula for u =
P
n cn z
n
in VMOgen (T),
Explicit description of the kernel of Γc when it is non trivial !
Extension of the result to non necessarily self adjoint
compact Hankel operators.
Method : construct a homeomorphism between VMOgen (T) and
the singular values of Γc and Γc̃ up to a choice of an element in
an infinite dimensional torus.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Explicit formula
Explicit formula for u =
P
n cn z
n
in VMOgen (T),
Explicit description of the kernel of Γc when it is non trivial !
Extension of the result to non necessarily self adjoint
compact Hankel operators.
Method : construct a homeomorphism between VMOgen (T) and
the singular values of Γc and Γc̃ up to a choice of an element in
an infinite dimensional torus.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Another formulation
L2+ (T)
= {u : u =
∞
X
ck e
ikθ
,
k=0
∞
X
|ck |2 < +∞ } ,
k=0
equipped with the symplectic form
Z
ω(u, v ) := Im
uv
T
dx
2π
Identify L2+ (T) with `2 (Z+ ) via the Fourier transform
u ∈ L2+ (T) 7→ (û(n))n∈Z+ .
Π : L2 (T) → L2+ (T) the Szegö projector.
Γc identifies to Hu Hu (f ) = Π(uf )
\
H
û(n) = cn , hp = f̂ (p).
u (f )(n) = (Γc (h))n ,
Γc̃ identifies to Ku = Hu Tz , Tz multiplication by z (shift on
Fourier coefficients).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Another formulation
L2+ (T)
= {u : u =
∞
X
ck e
ikθ
,
k=0
∞
X
|ck |2 < +∞ } ,
k=0
equipped with the symplectic form
Z
ω(u, v ) := Im
uv
T
dx
2π
Identify L2+ (T) with `2 (Z+ ) via the Fourier transform
u ∈ L2+ (T) 7→ (û(n))n∈Z+ .
Π : L2 (T) → L2+ (T) the Szegö projector.
Γc identifies to Hu Hu (f ) = Π(uf )
\
H
û(n) = cn , hp = f̂ (p).
u (f )(n) = (Γc (h))n ,
Γc̃ identifies to Ku = Hu Tz , Tz multiplication by z (shift on
Fourier coefficients).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Another formulation
L2+ (T)
= {u : u =
∞
X
ck e
ikθ
,
k=0
∞
X
|ck |2 < +∞ } ,
k=0
equipped with the symplectic form
Z
ω(u, v ) := Im
uv
T
dx
2π
Identify L2+ (T) with `2 (Z+ ) via the Fourier transform
u ∈ L2+ (T) 7→ (û(n))n∈Z+ .
Π : L2 (T) → L2+ (T) the Szegö projector.
Γc identifies to Hu Hu (f ) = Π(uf )
\
H
û(n) = cn , hp = f̂ (p).
u (f )(n) = (Γc (h))n ,
Γc̃ identifies to Ku = Hu Tz , Tz multiplication by z (shift on
Fourier coefficients).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Another formulation
L2+ (T)
= {u : u =
∞
X
ck e
ikθ
,
k=0
∞
X
|ck |2 < +∞ } ,
k=0
equipped with the symplectic form
Z
ω(u, v ) := Im
uv
T
dx
2π
Identify L2+ (T) with `2 (Z+ ) via the Fourier transform
u ∈ L2+ (T) 7→ (û(n))n∈Z+ .
Π : L2 (T) → L2+ (T) the Szegö projector.
Γc identifies to Hu Hu (f ) = Π(uf )
\
H
û(n) = cn , hp = f̂ (p).
u (f )(n) = (Γc (h))n ,
Γc̃ identifies to Ku = Hu Tz , Tz multiplication by z (shift on
Fourier coefficients).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Another formulation
L2+ (T)
= {u : u =
∞
X
ck e
ikθ
,
k=0
∞
X
|ck |2 < +∞ } ,
k=0
equipped with the symplectic form
Z
ω(u, v ) := Im
uv
T
dx
2π
Identify L2+ (T) with `2 (Z+ ) via the Fourier transform
u ∈ L2+ (T) 7→ (û(n))n∈Z+ .
Π : L2 (T) → L2+ (T) the Szegö projector.
Γc identifies to Hu Hu (f ) = Π(uf )
\
H
û(n) = cn , hp = f̂ (p).
u (f )(n) = (Γc (h))n ,
Γc̃ identifies to Ku = Hu Tz , Tz multiplication by z (shift on
Fourier coefficients).
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Action-angle coordinates
Key point : Hu Tz = Tz∗ Hu .
Ku2 = (Hu Tz )2 = Hu2 − (·|u)u.
u ∈ VMO so that Hu and Ku compact.
1
(ρ2j (u))j≥1 non zero eigenvalues of Hu2 ,
2
2 (u))
2
(σm
m≥1 non zero eigenvalues of Ku ,
ρ21 (u) ≥ σ12 (u) ≥ ρ22 (u) ≥ · · · ≥ ρ2N (u) ≥ σN2 (u) ≥ · · · → 0
VMOgen := VMO ∩ {ρ1 > σ1 > ρ2 > · · · > ρN > σN > · · · → 0}
dense subset of VMO.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Action angle
Antilinearity of Hu and of Ku :
orthonormal family (ej )j≥1 of Im(Hu )
orthonormal family (fm )m≥1 of Im(Ku ) such that
Hu (ej ) = ρj (u)ej , j ≥ 1; Ku (fm ) = σm (u)fm , m ≥ 1
(ej ), (fm ) defined up to a change of sign.
1
ϕj (u) := arg(1|ej )2 , j ≥ 1
2
θm (u) := arg(u|fm )2 , m ≥ 1
(well defined since (1|ej ) 6= 0 and (u|fm ) 6= 0)
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Homeomorphism
Let
Ξ := {(ζj )j≥1 ∈ CZ+ , |ζ1 | > |ζ2 | > |ζ3 | > |ζ4 | · · · > · · · → 0}.
Theorem
The map
χ := u 7→ ζ = ((ζ2j−1 = ρj e−iϕj )j≥1 , (ζ2j = σj e−iθj )j≥1 )
is a homeomorphism from VMOgen onto Ξ.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Explicit formula
If ζ is given in Ξ, then the Fourier coefficients of u are given by
û(n) = X .An Y ,
X = (νj ζ2j−1 )j≥1 , Y = (νj )j≥1 ,
where A = (Ajk )j,k ≥1 is the bounded operator on `2 defined by
Ajk =
∞
X
m=1
νj νk ζ2k−1 κ2m ζ2m
, j, k ≥ 1 ,
(|ζ2j−1 |2 − |ζ2m |2 )(|ζ2k−1 |2 − |ζ2m |2 )
with
νj2 :=
κ2m
|ζ2j |2
1−
|ζ2j−1 |2
2
!
Y
k6=j
2
:= |ζ2m−1 | − |ζ2m |
|ζ2j−1 |2 − |ζ2k |2
|ζ2j−1 |2 − |ζ2k−1 |2
− |ζ2`−1 |2
|ζ2m |2 − |ζ2` |2
Y |ζ
`6=m
Sandrine Grellier
2
2m |
!
,
,
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Compressed shift operator
S = Pu Tz compressed shift operator.
Pu orthogonal projector on Im(Hu ) = Ker(Hu )⊥ . In particular
Ku = Hu S.
ker Hu is stable under Tz , S = Pu Tz Pu .
u(z) =
∞
X
n
n
(u|z )z =
n=0
=
∞
X
∞
X
(u|Tzn (1))z n
n=0
(u|Pu Tzn (1))z n
n=0
=
∞
X
(u|S n Pu (1))z n .
n=0
Compute cn = (u|S n Pu (1)) in terms of ζ by writing u, Pu (1) in
the orthonormal basis (ej ) and the action of S on it.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Compressed shift operator
S = Pu Tz compressed shift operator.
Pu orthogonal projector on Im(Hu ) = Ker(Hu )⊥ . In particular
Ku = Hu S.
ker Hu is stable under Tz , S = Pu Tz Pu .
u(z) =
∞
X
n
n
(u|z )z =
n=0
=
∞
X
∞
X
(u|Tzn (1))z n
n=0
(u|Pu Tzn (1))z n
n=0
=
∞
X
(u|S n Pu (1))z n .
n=0
Compute cn = (u|S n Pu (1)) in terms of ζ by writing u, Pu (1) in
the orthonormal basis (ej ) and the action of S on it.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Trace formula
J(x) =
((I−xHu2 )−1 (1)|1)
∞
X
= 1+
x n J2n (u), J2n (u) = (Hu2n (1)|1).
n=1
Trace of (I − xHu2 )−1 − (I − xKu2 )−1 linked to
J 0 (x)
J(x)
Lemma
J(x) =
∞
Y
1 − xσj2 (u)
j=1
1 − xρ2j (u)
As
νj2
=
= 1+x
∞
X
ρ2j (u)|(1|ej )|2
1 − xρ2j (u)
j=1
1−
σj2
ρ2j
Y
ρ2j − σk2
k6=j
ρ2j − ρ2k
!
(
, x∈
/
1
2
ρj (u)
)
.
j≥1
!
we get |(1|ej )| = νj .
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Explicit formula
(1|ej )ej =
X
u = Hu (1) = Hu (Pu (1)) =
X
Pu (1) =
X
j
νj eiϕj /2 ej ,
j
ρj νj e−iϕj /2 ej .
j
Lemma
2 I)−1 (u) is an
The sequence (gm ) defined by gm = (Hu2 − σm
orthogonal basis of the range of Ku and
S(gm ) = σm eiθm hm
where
2 −1
hm = (Hu2 − σm
I) (1)
ζ given, explicit formula for u. So χ is one-to-one.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Explicit formula
(1|ej )ej =
X
u = Hu (1) = Hu (Pu (1)) =
X
Pu (1) =
X
j
νj eiϕj /2 ej ,
j
ρj νj e−iϕj /2 ej .
j
Lemma
2 I)−1 (u) is an
The sequence (gm ) defined by gm = (Hu2 − σm
orthogonal basis of the range of Ku and
S(gm ) = σm eiθm hm
where
2 −1
hm = (Hu2 − σm
I) (1)
ζ given, explicit formula for u. So χ is one-to-one.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Explicit formula
(1|ej )ej =
X
u = Hu (1) = Hu (Pu (1)) =
X
Pu (1) =
X
j
νj eiϕj /2 ej ,
j
ρj νj e−iϕj /2 ej .
j
Lemma
2 I)−1 (u) is an
The sequence (gm ) defined by gm = (Hu2 − σm
orthogonal basis of the range of Ku and
S(gm ) = σm eiθm hm
where
2 −1
hm = (Hu2 − σm
I) (1)
ζ given, explicit formula for u. So χ is one-to-one.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Explicit formula
(1|ej )ej =
X
u = Hu (1) = Hu (Pu (1)) =
X
Pu (1) =
X
j
νj eiϕj /2 ej ,
j
ρj νj e−iϕj /2 ej .
j
Lemma
2 I)−1 (u) is an
The sequence (gm ) defined by gm = (Hu2 − σm
orthogonal basis of the range of Ku and
S(gm ) = σm eiθm hm
where
2 −1
hm = (Hu2 − σm
I) (1)
ζ given, explicit formula for u. So χ is one-to-one.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Explicit formula
(1|ej )ej =
X
u = Hu (1) = Hu (Pu (1)) =
X
Pu (1) =
X
j
νj eiϕj /2 ej ,
j
ρj νj e−iϕj /2 ej .
j
Lemma
2 I)−1 (u) is an
The sequence (gm ) defined by gm = (Hu2 − σm
orthogonal basis of the range of Ku and
S(gm ) = σm eiθm hm
where
2 −1
hm = (Hu2 − σm
I) (1)
ζ given, explicit formula for u. So χ is one-to-one.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
The finite rank case
As Ku2 = Hu2 − (·|u)u, rank(H2u ) − 1 ≤ rank(K2u ) ≤ rank(H2u ).
V(2N) := {u, rank(H2u ) = N = rank(K2u )}.
Consider
V(2N)gen := {u ∈ V(2N), ρ1 > σ1 > · · · > ρN > σN > 0} and
ΞN := {ζ ∈ C2N ; |ζ1 | > |ζ2 | > · · · > |ζ2N−1 | > |ζ2N | > 0}.
Theorem (P. Gérard - S.G. 2011)
The map
χN : u 7→ ζ = ((ζ2j−1 = ρj e−iϕj )1≤j≤N , (ζ2j = σj e−iθj )1≤j≤N ) is a
symplectic diffeomorphism from (V(2N)gen , ω) onto (ΞN , ω 0 ),
2N
N
X
1 X
ω =
dζj ∧ dζ j =
ρj dρj ∧ dϕj + σj dσj ∧ dθj .
2i
0
j=1
j=1
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
The finite rank case
As Ku2 = Hu2 − (·|u)u, rank(H2u ) − 1 ≤ rank(K2u ) ≤ rank(H2u ).
V(2N) := {u, rank(H2u ) = N = rank(K2u )}.
Consider
V(2N)gen := {u ∈ V(2N), ρ1 > σ1 > · · · > ρN > σN > 0} and
ΞN := {ζ ∈ C2N ; |ζ1 | > |ζ2 | > · · · > |ζ2N−1 | > |ζ2N | > 0}.
Theorem (P. Gérard - S.G. 2011)
The map
χN : u 7→ ζ = ((ζ2j−1 = ρj e−iϕj )1≤j≤N , (ζ2j = σj e−iθj )1≤j≤N ) is a
symplectic diffeomorphism from (V(2N)gen , ω) onto (ΞN , ω 0 ),
2N
N
X
1 X
ω =
dζj ∧ dζ j =
ρj dρj ∧ dϕj + σj dσj ∧ dθj .
2i
0
j=1
j=1
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Local diffeomorphism
Hamiltonian vector field of F XF on M defined by
∀m ∈ M, ∀h ∈ Tm M, dF (m).h = ω(h, XF (m)) .
Poisson brackets of F and G is
{F , G} = dG.XF = ω(XF , XG ) .
ω0 =
X
ρj dρj ∧ dϕj + σj dσj ∧ dθj
j
is equivalent to commutation properties :
Proposition
{ρj , ρk } = {ρj , σk } = {σj , σk } = 0
{ρj , ϕk } = ρj δjk , {σj , ϕk } = 0, {ρj , θk } = 0, {σj , θk } = σj δjk ...
implies local diffeomorphism. Proof via the Szegö hierarchy.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Szegö hierarchy
Recall that J(x) = ((I − xHu2 )−1 (1)|1).
Theorem (Szegö hierarchy (P. Gérard- S.G. 2010))
If u is solution of Equation ∂t u = XJ(x) (u) then it satisfies
∂t Hu = [Bux , Hu ] , ∂t Ku = [Cux , Ku ]
with Bux et Cux skew-adjoint, explicit formulae.
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators
Inverse pb Action-angle Proof
Poisson brackets
Lemma
{J(x), ρj } = {J(x), σm } = 0,
1 xJ(x)
1 xJ(x)
, {J(x), θm } = −
{J(x), ϕj } =
.
2
2x
2 1 − ρj x
2 1 − σm
Two first commutations : Bux , Cux skew-adjoint so spectra of Hu
and of Ku invariant under XJ(x) .
Third and fourth : compute ėj and ḟm under XJ(x) .
Then J(x) =
N
Y
1 − σj2 x
j=1
1 − ρ2j x
so that {ρj , ϕk } = ρj δjk , {σj , ϕk } = 0...
Sandrine Grellier
Inverse Spectral problem for compact Hankel Operators