Two-sided and one-sided invertibility of Wiener-type functional operators with a 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 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 $$A={\sum }_{k\in \mathbb{Z}}{a}_k{U}_{\alpha }^k\text{where}\Vert A{\Vert }_W={\sum }_{k\in \mathbb{Z}}\Vert a_k{\Vert }_{L^{\infty }\left({\mathbb{R}}_{+}\right)}<\infty ,$$ 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 $t\in {\mathbb{R}}_{+}$.