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

Antonevich, A. B.; Ahmatova, A. A.; Makowska, J.

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 \(\alpha \).

Reviewer: Sophie Grivaux (Amiens)

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

Keywords:

spectrum of an operator; one-sided invertibility; essential spectrum; attractor; ergodic measure