Abstract. 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.

AMS Subject Classification (1991): 46A03, 46A04, 47A15, 47A99, 47L10

Keyword(s): locally convex space, algebraic operator, nilpotent operator, invariant subspace, reflexivity, hyporeflexivity

Received June 9, 2010, and in revised form October 14, 2010. (Registered under 38/2010.)