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 Izv. Akad. Nauk SSSR Ser. Mat., 1975, Volume 39, Issue 5, Pages 1130–1141 (Mi izv2082)

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.

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English version:
Mathematics of the USSR-Izvestiya, 1975, 9:5, 1069–1079

Bibliographic databases:

UDC: 513.88
MSC: Primary 47B50; Secondary 47H10

Citation: V. A. Khatskevich, “On fixed points of generalized linear-fractional transformations”, Izv. Akad. Nauk SSSR Ser. Mat., 39:5 (1975), 1130–1141; Math. USSR-Izv., 9:5 (1975), 1069–1079

Citation in format AMSBIB
\Bibitem{Kha75} \by V.~A.~Khatskevich \paper On fixed points of generalized linear-fractional transformations \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1975 \vol 39 \issue 5 \pages 1130--1141 \mathnet{http://mi.mathnet.ru/izv2082} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=405163} \zmath{https://zbmath.org/?q=an:0324.47025} \transl \jour Math. USSR-Izv. \yr 1975 \vol 9 \issue 5 \pages 1069--1079 \crossref{https://doi.org/10.1070/IM1975v009n05ABEH001508} 

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This publication is cited in the following articles:
1. V. A. Khatskevich, “An application of the contraction mapping principle in the theory of operators in an indefinite metric space”, Funct. Anal. Appl., 12:1 (1978), 70–71
2. A. V. Sobolev, V. A. Khatskevich, “Definite invariant subspaces and the structure of the spectrum of a focusing plus-operator”, Funct. Anal. Appl., 15:1 (1981), 69–70
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