= Weak-star fixed point theorem for an adjoint operator
{title2=$T(C)\subseteq C\Longrightarrow\operatorname{Fix}(T)\cap C\ne\varnothing$}
If $T=S^*$ for a <bounded operator> on a separable <Banach space>, and $C$ is a nonempty weak-star closed convex subset of the dual <unit ball> with $T(C)\subseteq C$, then $T$ has a fixed point in $C$. The <Cesàro averages> of the orbit remain in $C$, and $T$ minus the identity sends those averages to <vectors> of <norm> at most $2/n$. Weak-star <compactness> supplies a cluster point, and the adjoint action is weak-star continuous. The corresponding assertion with only weak closedness is false for the right shift on the probability simplex in $\ell^1$, the dual of $c_0$.
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