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.

Lomonosov's invariant subspace theorem

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. $S(M)\subset M$.

References

• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.