Operators generated by Morse-Smale mappings

Weighted shift operators $B$ in space $L^2(X,\mu )$ that are induced by Morse-Smale type of mappings are considered. A description of the properties of $B-\lambda I$ for $\lambda$ belonging to spectrum $\mathrm{\Sigma }(B)$ is given. In particular, there is the necessary and sufficient condition that $B-\lambda I$ be a one-sided invertible and the condition that set $Im(B-\lambda I)$ be non-closed. These conditions use a new notation: an oriented decomposition of oriented graph $G(X,\alpha )$ generated by mapping $\alpha$