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- W2000794976 abstract "A bounded linear operator T acting on a Banach space B is called weakly hypercyclic if there exists x∈B such that the orbit {Tnx:n=0,1,…} is weakly dense in B and T is called weakly supercyclic if there is x∈B for which the projective orbit {λTnx: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 ℓp(Z), 1⩽p<∞, is weakly supercyclic if and only if 2<p<∞ and that any weakly supercyclic weighted bilateral shift on ℓp(Z) for 1⩽p⩽2 is norm supercyclic. It is also shown that any weakly hypercyclic weighted bilateral shift on ℓp(Z) for 1⩽p<2 is norm hypercyclic, which answers a question of Chan and Sanders." @default.
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- W2000794976 date "2007-01-01" @default.
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- W2000794976 title "Non-sequential weak supercyclicity and hypercyclicity" @default.
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