Weak stability and quasistability

It is known that weak l-sequential supercyclicity implies weak quasistability, and it is still unknown whether weak l-sequential supercyclicity implies weak stability, much less whether weak supercyclicity implies weak stability (although it has been 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 weakly quasistable operator that is not weakly stable.