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

Theorem

Let $X$ be an infinite dimensional complex Banach space, $T \in ℬ (X)$ be compact and such that $T \neq 0$ . Further let $S \in ℬ (X)$ 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]

Citations

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References

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Lomonosov's invariant subspace theorem

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