Lomonosov's invariant subspace theorem explained

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

l{B}(X):= l{B}(X,X) be the space of bounded linear operators from some space X to itself. For an operator T\in l{B}(X) we call a closed subspace M\subset X, M ≠ \{ 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 l{B}(X) be compact and such that T ≠ 0 . Further let S\in l{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 .^[2]

Notes and References

1. Invariant subspaces for the family of operators which commute with a completely continuous operator. Victor I.. Lomonosov. Functional Analysis and Its Applications. 7. 213–214. 1973.
2. Book: Walter. Rudin. Functional Analysis. McGraw-Hill Science/Engineering/Math. 978-0070542365. 269-270.