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

Weak l-sequential supercyclicity and weak quasistability