Non-sequential weak supercyclicity and hypercyclicity

By Stanislav Shkarin. 2012

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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 { T n x : n = 0 , 1 , … } is weakly dense in B and T is called weakly supercyclic if there is x ∈ B for which the projective orbit { λ T n x : 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 M f ( z ) = z f ( z ) acting on L 2 ( μ ) is weakly supercyclic. It is not weakly sequentially supercyclic, since the projective orbit under M of each element in L 2 ( μ ) is weakly sequentially closed. This answers a question posed by Bayart and Matheron. It is proved that the bilateral shift on l p ( Z ) , 1 ⩽ p ∞ , is weakly supercyclic if and only if 2 p ∞ and that any weakly supercyclic weighted bilateral shift on l p ( Z ) for 1 ⩽ p ⩽ 2 is norm supercyclic. It is also shown that any weakly hypercyclic weighted bilateral shift on l p ( Z ) for 1 ⩽ p 2 is norm hypercyclic, which answers a question of Chan and Sanders.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f1a186a618aa19742486cc41f41910634e201e74