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]

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. $S(M)\subset M$ .^[2]

Citations

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

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.

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