A question in ${l^\infty }$

Question

Let ${l^\infty }$be the space of all real bounded functions $x$ on the positive integers. let $\tau $ be the translation operator defined on ${l^\infty }$ by the equation Let 'r be

$$(\tau x)(n) = x(n q- 1) $$

for $n = 1, 2, 3,\cdots$.

Why there exists a linear functional $\Psi $ on ${l^\infty }$ such that

• $\Psi \tau x = \Psi x$
• ${\rm{lim }}\inf x(n) \le \Psi x \le {\rm{lim}} \sup x(n)$

for every $x \in {l^\infty }$

Answer 1

Looks like a job for the Schauder fixed-point theorem.

EDIT: I'm assuming $q$ is an integer > 1, so $n \in \mathbb N \implies qn - 1 \in \mathbb N$. The unit ball $B$ of $(\ell^\infty)^*$ with the weak-* topology is compact. The intersection $K$ of the subsets $\{\psi \in B: \liminf x \le \psi(x) \le \limsup x \}$ is again compact and convex (and nonempty, by the Finite Intersection Property), and $\tau$ maps $K$ into itself. The Schauder fixed-point theorem provides a fixed point.