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 RussianâÂÂAmerican mathematician Victor Lomonosov.
Let be the space of bounded linear operators from some space to itself. For an operator we call a closed subspace an invariant subspace if , i.e. for every .
Let be an infinite dimensional complex Banach space, be compact and such that . Further let be an operator that commutes with . Then there exist an invariant subspace of the operator , i.e. .