Markov Kakutani fixed point theorem

Theorem [Markov - Kakutani] - Let $X$ be a topological vector space and $𝒯$ 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 $T\in 𝒯$. Then there is a point $x_0\in K$ such that $T{x}_0=x_0$ for all $T\in 𝒯$.