Let $K$ be the closed unit ball of some infinite dimensional Banach space, and let $H$ be an autohomeomorphism of $K$, having fixed points. Can $H/2$ be fixed point free ?

Also, let ${\mathcal{F}}$ := { $S\in\mbox{C}(K,K), \mbox{Fix}(S)\neq\textrm{Ø } $}.

Let $T$ in $\mbox{C}(K,K)$ such that $TS\in\mathcal{F}$ for all $S\in\mathcal{F}$ . Must $T$ be necessarily compact ?