Maps with separable dynamics and the spectral properties of the operators generated by them

Abstract

A map α of a space X into itself generates weighted shift operators B in function spaces on X. The spectral properties of such operators are intimately connected with the dynamics of α. It was known previ- ously that the spectrum of an operator depends only on the set of invariant ergodic measures for α. Conditions for the right invertibility of the operators B − λI are obtained when λ is a spectral value. The main result states that right invertibility is only possible when a nontrivial attractor exists.