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]

Citations

1. Lomonosov, Victor I. (1973). "Invariant subspaces for the family of operators which commute with a completely continuous operator". Functional Analysis and Its Applications. 7 (3): 213–214. doi:10.1007/BF01080698.
2. Rudin, Walter (1991). 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.