Non-sequential weak supercyclicity and hypercyclicity

A bounded linear operator T acting on a Banach space B is called weakly hypercyclic if there exists x ∈ B such that the orbit {T nx: n = 0, 1, . . .} is weakly dense in B and T is called weakly supercyclic if there is x ∈ B for which the projective orbit {λT nx: λ ∈ C, n = 0, 1, . . .} is weakly dense in B. If weak density is replaced by weak sequential density, then T is said to be weakly sequentially hypercyclic or supercyclic, respectively. It is shown that on a separable Hilbert space there are weakly supercyclic operators which are not weakly sequentially supercyclic. This is achieved by constructing a Borel probability measure μ on the unit circle for which the Fourier coefficients vanish at infinity and the multiplication operator Mf (z) = zf (z) acting on L2(μ) is weakly supercyclic. It is not weakly sequentially supercyclic, since the projective orbit under M of each element in L2(μ) is weakly sequentially closed. This answers a question posed by Bayart and Matheron. It is proved that the bilateral shift on p(Z), 1 p <∞, is weakly supercyclic if and only if 2