On the kernel of a singular integral operator with shift - Marreiros, Rui (2018)
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.