Weak l-sequential supercyclicity and weak quasistability

It is known that supercyclicity implies strong stability. It is not known whether weak l-sequential supercyclicity implies weak stability. In this paper we prove that weak l-sequential supercyclicity implies weak quasistability. Corollaries concerning the characterisation of (i) weakly l-sequentially supercyclic vectors that are not (strongly) supercyclic, and (ii) weakly l-sequentially supercyclic isometries, are also proved.