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Proceedings of the American Mathematical Society

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A fixed-point theorem for certain operator valued maps

Authors: D. R. Brown and M. J. O’Malley
Journal: Proc. Amer. Math. Soc. 74 (1979), 254-258
MSC: Primary 47H10
MathSciNet review: 524296
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Abstract: Let H be a real Hilbert space, and let $ {B_1}(H)$ denote the space of symmetric, bounded operators on H which have numerical range in [0, 1], topologized by the strong operator topology, and let L be a strongly continuous function on H into $ {B_1}(H)$. In this paper, methods are given to locate all $ z \in H$ which are fixed points of L in the sense that $ L(z)z = z$.

In particular, if $ w \in H$ and if $ \alpha $ and $ \beta $ are fixed positive rational numbers with $ \alpha \in [\tfrac{1}{2},\infty )$, a decreasing sequence of elements of $ {B_1}(H)$ is recursively defined, and converges to $ Q \in {B_1}(H)$. If $ \alpha > \tfrac{1}{2}$, then Q is idempotent and $ z = Qw$ is a fixed point of L, and if $ \alpha = \tfrac{1}{2},\beta \geqslant \tfrac{1}{2}$, then $ z = {Q^\beta }w$ is a fixed point of L.

References [Enhancements On Off] (What's this?)

  • [1] John L. Kelley, General topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. MR 0070144
  • [2] J. W. Neuberger, Projection methods for linear and nonlinear systems of partial differential equations, Ordinary and partial differential equations (Proc. Fourth Conf., Univ. Dundee, Dundee, 1976) Springer, Berlin, 1976, pp. 341–349. Lecture Notes in Math., Vol. 564. MR 0601044
  • [3] Martin Schechter, Principles of functional analysis, Academic Press, New York-London, 1971. MR 0445263

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Keywords: Positive operators, fixed points
Article copyright: © Copyright 1979 American Mathematical Society