Communications of the
Korean Mathematical Society CKMS

In the present paper, we show that a continuous linear operator $T$ on a Frechet space satisfies the Hypercyclic Criterion with respect to a syndetic sequence must satisfy the Kitai Criterion. On the other hand, an operator, hereditarily hypercyclic with respect to a syndetic sequence must be mixing. We also construct weighted shift operators satisfying the Hypercyclicity Criterion which do not satisfy the Kitai Criterion. In other words, hereditarily hypercyclic operators without being mixing.

Keywords: hypercyclic operator, hereditarily criterion, mixing operator

MSC numbers: Primary 47A16; Secondary 47B38