On spectral properties of weighted shift operators generated by linear mappings with perron’s conditions

Weighted shift operators $B$ in the space $L_2 (\mathbb{C}^m )$ generated by a linear mapping $A\colon\mathbb{C}^m \to \mathbb{C}^m$ are considered. A description of properties of $B - \lambda I$ for $\lambda$ belonging to spectrum $\sum(B)$ is given. In particular, the necessary and sufficient condition for $B - \lambda I$ to be one-sided invertible is obtained.