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

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 ?

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 ?

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Ady
Ady
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  • 24
  • 32

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

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