Lomonosov's invariant subspace theorem explained
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
be the space of bounded linear operators from some space
to itself. For an operator
we call a closed subspace
an invariant subspace if
, i.e.
for every
.
Theorem
Let
be an infinite dimensional complex Banach space,
be
compact and such that
. Further let
be an operator that commutes with
. Then there exist an invariant subspace
of the operator
, i.e.
.
[2] Notes and References
- Invariant subspaces for the family of operators which commute with a completely continuous operator. Victor I.. Lomonosov. Functional Analysis and Its Applications. 7. 213–214. 1973.
- Book: Walter. Rudin. Functional Analysis. McGraw-Hill Science/Engineering/Math. 978-0070542365. 269-270.