Lomonosov's invariant subspace theorem is a mathematical theorem from functional analysis concerning the existence of invariant subspaces of a

linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

on some complex

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

. The theorem was proved in 1973 by the Russian–American mathematician Victor Lomonosov.

Lomonosov's invariant subspace theorem

Notation and terminology

Let

\mathcal(X):=\mathcal(X,X)

be the space of bounded linear operators from some space

X

to itself. For an operator

T\in\mathcal(X)

we call a closed subspace

M\subset X,\;M\neq \

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(X)

be compact and such that

T\neq 0

. Further let

S\in\mathcal(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

Citations

References

* {{Functional Analysis Banach spaces category:Functional analysis category:Operator theory Theorems in functional analysis