## Maps with separable dynamics and the spectral properties of the operators generated by them.(English. Russian original)Zbl 1334.47034

Sb. Math. 206, No. 3, 341-369 (2015); translation from Mat. Sb. 206, No. 3, 3-34 (2015).
Let $$\alpha :X\longrightarrow X$$ be an invertible continuous map from a compact metric space $$X$$ into itself. The present paper is concerned with the study of spectral properties of weighted shift operators associated to $$\alpha$$ on the spaces $$L^{2}(X,\mu )$$, where $$\mu$$ is a fixed Borel measure on $$X$$. These weighted shift operators are defined by setting $$B_{u}(x)=a_{0}(x)u(\alpha (x))$$ for every function $$u\in L^{2}(X,\mu )$$. Here, $$a_{0}$$ is a weight function, which is such that $$B$$ is a bounded operator on $$L^{2}(X,\mu )$$.
The spectral properties of $$B$$ are studied in connection with the dynamics of the map $$\alpha$$. It is shown that there exists a weight function such that $$B-\lambda$$ is right invertible for some spectral value $$\lambda$$ of $$B$$ if and only if the map $$\alpha$$ has a property called separable dynamics, which is equivalent to the existence of a non-trivial attractor for the map $$\alpha$$.

### MSC:

 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A10 Spectrum, resolvent 47C15 Linear operators in $$C^*$$- or von Neumann algebras 47A35 Ergodic theory of linear operators
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