Markov Kakutani fixed point theorem

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Theorem [Markov - Kakutani] - Let $X$ be a topological vector space and $\mathcal{T}$ a commuting family of continuous linear operators in $X$ . Suppose $K$ is a compact convex subset of $X$ such that

T(K)\subseteq K

for every $T\in\mathcal{T}$ . Then there is a point $x_{0}\in K$ such that $Tx_{0}=x_{0}$ for all $T\in\mathcal{T}$ .

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