Weighted translation operators generated by mappings with saddle points: a model class

A particular class of weighted translation operators $B$ generated by mappings with saddle points are considered. For $\lambda$ belonging to the spectrum of the operator $B$, a description of properties of the operator $B-\lambda I$ is found. In particular, necessary and sufficient conditions of one-side invertibility are found. It follows from the obtained results that weighted translation operators generated by mappings with saddle points have principally different spectral properties compared to weighted translation operators generated by mappings without saddle points (investigated earlier).
It is proved that the operator $B-I$ is one-side invertible if and only if a certain property of a linear extension associated with the operator $B$ holds.