On fixed points of generalized linear-fractional transformations

V. A. Khatskevich

Izvestiya. Mathematics , Tome 9 (1975) no. 5, pp. 1069-1079

Résumé

We study the fixed points of the generalized linear-fractional transformation $F_A$, induced by the plus-operator $A$, of the operator unit ball $\mathscr K_+$ into $\mathscr K_+$. In particular, for a linear-fractional transformation $F_A$ which maps $\mathscr K_+$ into its interior $\mathscr K_+^0$ we prove that if $F_A$ has a fixed point then the latter is unique. If, on the other hand, $F_A$ maps $\mathscr K_+$ onto $\mathscr K_+$, then, provided $F_A$ has a fixed point in $\mathscr K_+^0$, the following alternative is valid: 1) either this is the only fixed point of $F_A$ in $\mathscr K_+$, 2) or $F_A$ has a continuum of fixed points in the interior of $\mathscr K_+$ and at least two fixed points on the boundary $S_+$ of $\mathscr K_+$. In the intermediate case where $F_A(\mathscr K_+)\ne\mathscr K_+$ but $F_A(\mathscr K_+)\cap S_+\ne\varnothing$ we give an example of a linear-fractional transformation $F_A$ that has two fixed points: one in $\mathscr K_+^0$ and one on $S_+$. Bibliography: 11 titles.