  This paper considers weak supercyclicity for bounded linear operators on a
normed space. On the one hand, weak supercyclicity is investigated for classes
of Hilbert-space operators: (i) self-adjoint operators are not weakly
supercyclic, (ii) diagonalizable operators are not weakly l-sequentially
supercyclic, and (iii) weak l-sequential supercyclicity is preserved between a
unitary operator and its adjoint. On the other hand, weak supercyclicity is
investigated for classes of normed-space operators: (iv) the point spectrum of
the normed-space adjoint of a power bounded supercyclic operator is either
empty or is a singleton in the open unit disk, (v) weak l-sequential
supercyclicity coincides with supercyclicity for compact operators, and (vi)
every compact weakly l-sequentially supercyclic operator is quasinilpotent.
