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We consider the following Hamiltonian equation on the Hardy space on the circle,where is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating...
We compute the essential Taylor spectrum of a tuple of analytic Toeplitz operators on , where D is a strictly pseudoconvex domain. We also provide specific formulas for the index of provided that is a compact subset of D.
Let be a positive Borel measure on the complex plane and let with . We study the generalized Toeplitz operators on the Fock space . We prove that is bounded (or compact) on if and only if is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for to be in the Schatten -class for .
In this talk we explain a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy . This permits a detailed study of the spectrum in various asymptotic regions of the parameters , and gives improvements and new proofs for many of the results in the field. In the completely resonant...
We consider the solution operator to the -operator restricted to forms with coefficients in . Here denotes -forms with coefficients in , is the corresponding -space and is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula to . This solution operator will have the property . As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators...
Let be an inner function and be the corresponding model space. For an inner function , the subspace is an invariant subspace of the unilateral shift operator on . In this article, using the structure of a Toeplitz kernel , we study the intersection by properties of inner functions and
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