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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.
UDC:
513.88
MSC: Primary
47B50; Secondary
47H10 Received: 14.01.1974
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