## 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 isometric shift operator acting on the Lebesgue space ${L}^{p}\left({\mathbb{R}}_{+}\right)$ with $p\in [1,\infty ]$ by the rule ${U}_{\alpha}f=\left(\alpha \text{'}{)}^{1/p}\right(f\circ \alpha )$. We establish criteria of the two-sided and one-sided invertibility of functional operators of the form $$A={\sum}_{k\in \mathbb{Z}}{a}_{k}{U}_{\alpha}^{k}\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}\Vert A{\Vert}_{W}={\sum}_{k\in \mathbb{Z}}\Vert {a}_{k}{\Vert}_{{L}^{\infty}\left({\mathbb{R}}_{+}\right)}<\infty ,$$ on the spaces ${L}^{p}\left({\mathbb{R}}_{+}\right)$ under the assumptions that the functions $log\alpha \text{'}$ and ${a}_{k}$ for all $k\in \mathbb{Z}$ are bounded and continuous on ${\mathbb{R}}_{+}$ and may have slowly oscillating discontinuities at $0$ and $\infty $. The unital Banach algebra ${\mathfrak{A}}_{W}$ of such operators is inverse-closed: if $A\in {\mathfrak{A}}_{W}$ is invertible on ${L}^{p}\left({\mathbb{R}}_{+}\right)$ for $p\in [1,\infty ]$, then ${A}^{-1}\in {\mathfrak{A}}_{W}$. Obtained criteria are of two types: in terms of the two-sided or one-sided invertibility of so-called *discrete operators* on the spaces ${l}^{p}$ and in terms of conditions related to the fixed points of $\alpha $ and the orbits $\left\{{\alpha}^{n}\right(t):n\in \mathbb{Z}\}$ of points $t\in {\mathbb{R}}_{+}$.

## Citation

Gustavo Fernández-Torres. Yuri Karlovich. "Two-sided and one-sided invertibility of Wiener-type functional operators with a shift and slowly oscillating data." Banach J. Math. Anal. 11 (3) 554 - 590, July 2017. https://doi.org/10.1215/17358787-2017-0006

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