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

A map $\alpha $ 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 $\alpha$. It was known previously that the spectrum of an operator depends only on the set of invariant ergodic measures for $\alpha$. Conditions for the right invertibility of the operators $B-\lambda I$ are obtained when $\lambda$ is a spectral value. The main result states that right invertibility is only possible when a nontrivial attractor exists.
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