  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.
