Koenigs Problem and Extreme Fixed Points

This note continues some previous studies by the authors. We consider a linear-fractional mapping $\mathcal{F}_A\colon\mathcal{K}\to\mathcal{K}$ generated by a triangular operator, where $\mathcal{K}$ is the unit operator ball and the fixed point $C$ of the extension of $\mathcal{F}_A$ to $\overline{\mathcal{K}}$ is either an isometry or a coisometry. Under some natural restrictions on one of the diagonal entries of the operator matrix $A$, the structure of the other diagonal entry is investigated completely. It is shown that generally $C$ cannot be replaced in all these considerations by an arbitrary point of the unit sphere. Some special cases are studied in which this is nevertheless possible.
In conclusion, the Koenigs embedding property of the mappings under study is proved with the use of the results announced in this paper.