On the study of the dimension of the kernel of singular integral operators with non-carleman shift using mathematica software - Marreiros, Rui (2012)
We consider the singular integral operator T=I-cUP_+ : L_2 (T)→L_2 (T), with a non- Carleman shift, where I is the identity operator, c∈C(T) is a continuous matrix function on the unit circle T , U is the isometric shift operator and P_+ is the Cauchy projector. It is supposed that the shift has a finite set of fixed points and the modulus of the function c(t) at the fixed points of the shift is less than one. Under these conditions, an estimate for the dimension of the kernel of the operator T , is obtained. We consider some examples to illustrate and show that the obtained estimate, in a certain sense, is sharp.