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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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