Common Invariant Subspaces from Small Commutators

We study the following question: suppose that A and B are two algebras of complex n×n matrices such that the ring commutator [A, B] = AB−BA is “small” for each A ∈ A and B ∈ B. Does this imply that A and B have a common non-trivial invariant subspace? This question is motivated by a series of papers studying the structure of “almost-commutative” algebras and, more generally, semigroups. A simple example shows that in general the answer is no: it may happen that the algebra B is equal to the commutant A′ of A and the two algebras do not share an invariant subspace. We characterize all such algebras: if a matrix algebra A does not share invariant subspaces with its commutant, then it must be similar to an amplification of a full matrix algbera. Then we show that if A and B are two algebras such rank [A, B] 6 1 for all A ∈ A and B ∈ B and the rank-one is achieved, then A and B have a common invariant subspace. A number of partial results about linear spaces (rather than algebras) of matrices, as well as the condition that [A, B] is always nilpotent, are also discussed.