On the Kernel of some one-dimensional singular integral operators with shift

An estimate for the dimension of the kernel of the singular integral operator with shift (I+ Sigma(n)/j=1 aj(t)U-j) P+ +P- : L-2(R) -> L-2(R) is obtained, where P +/- are the Cauchy projectors, (U phi)(t) = phi(t + h), h epsilon R+, is the shift operator and a(j)(t) are continuous functions on the one point compactification of R. The roots of the polynomial 1 +Sigma(n)/j=1 a(j) (infinity)n(j) are assumed to belong all simultaneously either to the interior of the unit circle or to its exterior.