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

Abstract

Let Γ be an oriented Jordan smooth curve and α a diffeomorphism of Γ onto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertibility of the binomial functional operator $$A = aI - bW$$ wherea andb are continuous functions,I is the identity operator,W is the shift operator,Wf=foα, on a reflexive rearrangement-invariant spaceX(Γ) with Boyd indices α X , β X and Zippin indicesp x,q x satisfying inequalities $$0< \alpha x = px \leqslant qx = \beta x< 1$$