A fixed-point theorem for certain operator valued maps

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.