Hankel operators and integrable systems

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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


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Langue English
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Hankel operators and integrable systems
PatrickG´erard
Univ.Paris-Sud11,LaboratoiredeMathe´matiquesdOrsay,CNRS,UMR8628 and Institut Universitaire de France
fromajointworkwithSandrineGrellier(Orl´eans) ´ Ecole d’hiver “Dynamique et EDP”, ´ SaintEtiennedeTin´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)