Some properties of algebraic operators on locally convex spaces

We investigate some properties of an algebraic operator A on a general vector space X and especially in the case when X is a locally convex space. We prove that A is always hyporeflexive and that it is reflexive if its minimal polynomial is simple. Moreover, we show that this condition is necessary and sufficient for the reflexivity of the commutant of A. We also show that the second commutant of A is equal to the algebra generated by A and the identity operator. In the last section we prove that every locally algebraic operator acting on a Fréchet space is algebraic, and that an operator which is a finite rank perturbation of an algebraic operator is again algebraic.