fa.functional analysis - On the failure of the infinite dimensional Brouwer Theorem

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 ${mathcal{F}}$ := { $Sinmbox{C}(K,K), mbox{Fix}(S)neqtextrm{Ø } $}.

Let $T$ in $mbox{C}(K,K)$ such that $TSinmathcal{F}$ for all $Sinmathcal{F}$
. Must $T$ be necessarily compact ?