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.^[1]

Lomonosov's invariant subspace theorem

Notation and terminology

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

Theorem

Let ${\displaystyle X}$ be an infinite dimensional complex Banach space, ${\displaystyle T\in {\mathcal {B}}(X)}$ be compact and such that ${\displaystyle T\neq 0}$. Further let ${\displaystyle S\in {\mathcal {B}}(X)}$ be an operator that commutes with ${\displaystyle T}$. Then there exist an invariant subspace ${\displaystyle M}$ of the operator ${\displaystyle S}$, i.e. ${\displaystyle S(M)\subset M}$.^[2]