One-Sided Invertibility Criteria for Binomial Functional Operators with Shift and Slowly Oscillating Data

Abstract

Let \({\alpha}\) be an orientation-preserving homeomorphism of \({[0,\infty]}\) onto itself with only two fixed points at 0 and \({\infty}\), whose restriction to \({\mathbb{R}_+=(0,\infty)}\) is a diffeomorphism, and let \({U_\alpha}\) be the corresponding isometric shift operator acting on the Lebesgue space \({L^p(\mathbb{R}_+)}\) by the rule \({U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)}\). We prove criteria for the one-sided invertibility of the binomial functional operator \({aI-bU_\alpha}\) on the spaces \({L^p(\mathbb{R}_+)}\), \({p\in(1,\infty)}\), under the assumptions that a, b and \({\alpha'}\) are bounded and continuous on \({\mathbb{R}_+}\) and may have slowly oscillating discontinuities at 0 and \({\infty}\).