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This article is cited in 3 scientific papers (total in 3 papers)
On fixed points of generalized linear-fractional transformations
V. A. Khatskevich
Abstract:
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.
Received: 14.01.1974
Citation:
V. A. Khatskevich, “On fixed points of generalized linear-fractional transformations”, Math. USSR-Izv., 9:5 (1975), 1069–1079
Linking options:
https://www.mathnet.ru/eng/im2082https://doi.org/10.1070/IM1975v009n05ABEH001508 https://www.mathnet.ru/eng/im/v39/i5/p1130
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