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 $$ \left( {I + \sum\limits_{j = 1}^n {a_j (t)U^j } } \right)P_ + + P_ - :L_2 (\mathbb{R}) \to L_2 (\mathbb{R}) $$ is obtained, where P± are the Cauchy projectors, (U ψ)(t) = ψ(t+h), h ∈ ℝ+, is the shift operator and aj(t) are continuous functions on the one point compactification of ℝ. The roots of the polynomial $$ 1 + \sum\limits_{j = 1}^n {a_j (\infty )\eta ^j } $$ are assumed to belong all simultaneously either to the interior of the unit circle or to its exterior. [ABSTRACT FROM AUTHOR]