One-sided invertibility of binomial functional operators with a shift in rearrangement-invariant spaces

Abstract

Let $\Gamma$ be an oriented Jordan smooth curve and $\alpha$ be a diffeomorphism of $\Gamma$ onto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertiblity of the binomial functional operator \[ A=aI-bW \] where $a$ and $b$ are continuous functions, $I$ is the identity operator, $W$ is the shift operator $Wf=f\circ \alpha$, in a reflexive rearrangement-invariant space $X(\Gamma )$ with Boyd indices ${\alpha }_X,{\beta }_X$ and Zippin indices $p_X,q_X$ satisfying inequalities \[ 0<α_X=p_X\le q_X=q_X<1. \]