Weak Stability and Quasistability

It is known that weak l-sequential supercyclicity implies weak quasistability, and it is still unknown weather weak l-sequential supercyclicity implies weak stability, much less whether weak supercyclicity implies weak stability (although it is known for a long time that strong supercyclicity implies strong stability). It is shown that weak l-sequential supercyclicity implies weak stability under the assumption of boundedly spaced subsequences. It is given an example of a power bounded quasistable operator which is not weakly stable.