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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
DOI: https://doi.org/10.1090/S0002-9939-1979-0524296-X
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, Van Nostrand, Princeton, N. J., 1955. MR 0070144 (16:1136c)
  • [2] John Neuberger, Projection methods for linear and nonlinear systems of partial differential equations, Lecture Notes in Math., Springer, Berlin (to appear). MR 0601044 (58:29129)
  • [3] Martin Schechter, Principles of functional analysis, Academic Press, New York, 1971. MR 0445263 (56:3607)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0524296-X
Keywords: Positive operators, fixed points
Article copyright: © Copyright 1979 American Mathematical Society

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