Lomonosov's invariant subspace theorem

Lomonosov's invariant subspace theorem is a mathematical theorem from functional analysis concerning the existence of invariant subspaces of a linear operator on some complex Banach space. The theorem was proved in 1973 by the Russian–American mathematician Victor Lomonosov.

Notation and terminology
Let $$\mathcal{B}(X):=\mathcal{B}(X,X)$$ be the space of bounded linear operators from some space $$X$$ to itself. For an operator $$T\in\mathcal{B}(X)$$ we call a closed subspace $$M\subset X,\;M\neq \{0\}$$ an invariant subspace if $$T(M)\subset M$$, i.e. $$Tx\in M$$ for every $$x\in M$$.

Theorem
Let $$X$$ be an infinite dimensional complex Banach space, $$T\in\mathcal{B}(X)$$ be compact and such that $$T\neq 0$$. Further let $$S\in\mathcal{B}(X)$$ be an operator that commutes with $$T$$. Then there exist an invariant subspace $$M$$ of the operator $$S$$, i.e. $$S(M)\subset M$$.