Title of article :
CMV matrices with asymptotically constant
coefficients. Szegö–Blaschke class, scattering theory
Author/Authors :
F. Peherstorfer، نويسنده ,
Issue Information :
روزنامه با شماره پیاپی سال 2009
Abstract :
We develop a modern extended scattering theory for CMV matrices with asymptotically constant
Verblunsky coefficients. We show that the traditional (Faddeev–Marchenko) condition is too restrictive
to define the class of CMV matrices for which there exists a unique scattering representation. The main
results are: (1) the class of twosided CMV matrices acting in l2, whose spectral density satisfies the Szegö
condition and whose point spectrum the Blaschke condition, corresponds precisely to the class where the
scattering problem can be posed and solved. That is, to a given CMV matrix of this class, one can associate
the scattering data and the FM space. The CMV matrix corresponds to the multiplication operator in this
space, and the orthonormal basis in it (corresponding to the standard basis in l2) behaves asymptotically as
the basis associated with the free system. (2) From the point of view of the scattering problem, the most
natural class of CMV matrices is that one in which (a) the scattering data determine the matrix uniquely
and (b) the associated Gelfand–Levitan–Marchenko transformation operators are bounded. Necessary and
sufficient conditions for this class can be given in terms of an A2 kind condition for the density of the
absolutely continuous spectrum and a Carleson kind condition for the discrete spectrum. Similar conditions
close to the optimal ones are given directly in terms of the scattering data.
© 2009 Elsevier Inc. All rights reserved.
Keywords :
scattering theory , Carleson condition , Schur functions , Verblunskycoefficients , A2 condition , CMV and Jacobi matrices
Journal title :
Journal of Functional Analysis
Journal title :
Journal of Functional Analysis