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.{{cite journal|title=Invariant subspaces for the family of operators which commute with a completely continuous operator|first1=Victor I.|last1=Lomonosov|journal=Functional Analysis and Its Applications|volume=7|pages=213–214|date=1973|issue=3 |doi=10.1007/BF01080698 }}

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. $S(M)\subset M$ .{{cite book|first1=Walter|last1=Rudin|title=Functional Analysis|date=1991 |publisher=McGraw-Hill Science/Engineering/Math|isbn=978-0070542365|page=269-270}}

Citations

References

{{Rudin Walter Functional Analysis|edition=2}}

Category:Banach spaces

category:Functional analysis

category:Operator theory

Category:Theorems in functional analysis