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 ${\infty}$.