On the kernel of a singular integral operator with shift

Marreiros, Rui

http://hdl.handle.net/10400.1/11201

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Authors

Marreiros, Rui

Abstract(s)

Some estimates for the dimension of the kernel of the singular integral operator I - cUP(+) : L-p(n)(T) -> L-p(n)(T), p is an element of (1, infinity), with a non-Carleman shift are obtained, where P+ is the Cauchy projector, U is an isometric shift operator and c(t) is a continuous matrix function on the unit circle T. It is supposed that the shift has a finite set of fixed points and all the eigenvalues of the matrix c(t) at the fixed points, simultaneously belong either to the interior of the unit circle T or to its exterior. The case of an operator with a general shift is also considered. Some relations between those estimates and the resolvent set of the operator cU are pointed out.