On the orbit of invariant subspaces of linear operators in finite-dimensional spaces (new proof of a Halmos's result)

Abstract

P.R. Halmos proved that for a linear operator A over a finite-dimensional complex vector space E, every A-invariant subspace of E is the range of a commutant of A. His proof was based on a generalization of the concept of eigenvector. In this note, we give an invariant proof of this Halmos's theorem.