Hankel operators and integrable systems

Patrick Gérard

Description

Hankel operators and integrable systems Patrick Gerard Univ. Paris-Sud 11, Laboratoire de Mathematiques d'Orsay, CNRS, UMR 8628 and Institut Universitaire de France from a jointwork with Sandrine Grellier (Orleans) Ecole d'hiver “Dynamique et EDP”, Saint–Etienne de Tinee, February 7-11, 2011

• iff ?f ?


• real valued


• finite rank iff


• ecole d'hiver


• saint–etienne de tinee

Sujets

• Institut universitaire de France
• Grellier
• Gerard
• Schmidt

Informations

• Publié par: pefav
• Nombre de lectures: 21
• Langue: English

Extrait

Hankel operators and integrable systems

PatrickG´erard

Univ.Paris-Sud11,LaboratoiredeMathe´matiquesd’Orsay,CNRS,UMR8628 and Institut Universitaire de France

fromajointworkwithSandrineGrellier(Orl´eans) ´ Ecole d’hiver “Dynamique et EDP”, ´ Saint–EtiennedeTin´ee,February7-11,2011

1.

Inverse

sp

ectral

problems

for

Hankel

op

erators

Hankel operators A Hankel operator is an operator on ` 2 ( Z + ) of the form

∞ Γ c ( x ) = y , y n = X c n + p x p , p =0

where c = ( c n ) is a sequence of complex numbers with convenient behaviour at infinity. Various properties of Γ c can be read on the associated Fourier series

∞ u ( e i θ ) = X c n e in θ , θ ∈ T := R / 2 π Z , n =0

or the corresponding holomorphic function on the unit disc

∞ u ( z ) = X c n z n , | z | < 1 . n =0

Correspondences

• Γ c is finite rank iff u ( z ) is rational (Kronecker, 1881)

• Γ c is Hilbert Schmidt iff u ∈ H 1 / 2 ( T )

ˆ • Γ c is bounded iff ∃ f ∈ L ∞ ( T ) : c n = f ( n ) , n ≥ 0 (Nehari, 1957) or iff u ∈ BMO ( T )

ˆ • Γ c is compact iff ∃ f ∈ C ( T ) : c n = f ( n ) , n ≥ 0 (Hartman, 1958) or iff u ∈ VMO ( T )

1 / p p , p ( T ), 0 < p < ∞ ,

• Γ c is in the Schatten class S p iff u ∈ B p , p (Peller, Semmes, 1984)