Criteria for one-sided invertibility of a functional operator in rearrangement-invariant spaces of fundamental type

Let gamma be a simple open smooth curve and alpha be an orientation-preserv ing diffeomorphism of gamma onto itself which has only two fixed points. Cr iteria for one-sided invertibility of the functional operator A = aI - bW, where a and b are continuous functions, I is the identity operator, W is th e shift operator: (Wf) (t) = f[alpha (t)], in a reflexive rearrangement-inv ariant space of fundamental type X(gamma) with nontrivial Boyd indices, are obtained.