Two-sided and one-sided invertibility of Wiener-type functional operators with a shift and slowly oscillating data

Abstract

Let α be an orientation-preserving homeomorphism of [0,∞] onto itself with only two fixed points at 0 and ∞, whose restriction to R+=(0,∞) is a diffeomorphism, and let Uα be the isometric shift operator acting on the Lebesgue space Lp(R+) with p∈[1,∞] by the rule Uαf=(α')1/p(f∘α). We establish criteria of the two-sided and one-sided invertibility of functional operators of the form A=∑k∈ZakUαkwhere‖A‖W=∑k∈Z‖ak‖L∞(R+)<∞, on the spaces Lp(R+) under the assumptions that the functions logα' and ak for all k∈Z are bounded and continuous on R+ and may have slowly oscillating discontinuities at 0 and ∞. The unital Banach algebra AW of such operators is inverse-closed: if A∈AW is invertible on Lp(R+) for p∈[1,∞], then A−1∈AW. Obtained criteria are of two types: in terms of the two-sided or one-sided invertibility of so-called discrete operators on the spaces lp and in terms of conditions related to the fixed points of α and the orbits {αn(t):n∈Z} of points t∈R+.