Every infinite dimensional separable non-normable Fre chet space admits a continuous hypercyclic operator. A large class of separable countable inductive limits of Banach spaces with the same property is given, but an example of a separable complete inductive limit of Banach spaces which admits no hypercyclic operator is provided. It is also proved that no compact operator on a locally convex space is hypercyclic.
1998 Academic Press
In the article [1] S. Ansari solved a long-standing problem of Rolewicz showing that every infinite dimensional separable Banach space admits a hypercyclic operator. This result was obtained independently by L. . An operator T on a locally convex space E is called hypercyclic if Orb(T, x) :=[x, Tx, T 2 x, ...] is dense in E for some x # E. In this case x is a hypercyclic vector for T. A corollary of the main result of [1] asserts that every separable Fre chet space (i.e. complete metrizable locally convex space) admits a hypercyclic operator. Unfortunately the proof, which is correct for Banach spaces, contains a gap in the case of nonnormable Fre chet spaces. Indeed, the proof is based on Remark 1 (4) which is false, since it is very easy to show that the dual of every nonnormable Fre chet space E contains sequences (u n ) n such that (: n u n ) n is not equicontinuous in E$ for every sequence (: n ) n of strictly positive numbers. See, e.g., [14, 3.1.4, p. 431].
Hypercyclicity of continuous linear operators on non-normable Fre chet spaces has been considered by several authors like Gethner and Shapiro [8], Godefroy and Shapiro [9] among others. In [9] the authors study the hypercyclicity of partial differential operators with constant coefficients on Fre chet spaces with a continuous norm as H(C N ) or without a continuous norm as C (R N ). Cyclic and hypercyclic vectors are of importance in connection with invariant subspaces. An operator lacks closed nontrivial invariant subspaces (respectively, subsets) if and only if every non-zero article no.