2012/10/11 by Stanislav Shkarin, Shkarin, Stanislav · 1 citation
Mathematics · #47A16 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #math.DS #math.FA #msc:47A16
paper · pdf · doi:10.48550/arxiv.1210.3191
arxiv created 2012/10/11 · arxiv updated 2012/10/12
We prove a new criterion of weak hypercyclicity of a bounded linear operator on a Banach space. Applying this criterion, we solve few open questions. Namely, we show that if G is a region of \C bounded by a smooth Jordan curve Γ such that G does not meet the unit ball but Γ intersects the unit circle in a non-trivial arc, then M^* is a weakly hypercyclic operator on H2(G), where M is the multiplication by the argument operator Mf(z)=zf(z). We also prove that if g is a non-constant function from the Hardy space H^∞(\D) on the unit disk \D such that g(\D)∩\D=\varnothing and the set \z∈\C:|z|=1, |g(z)|=1\ is a subset of the unit circle \T of positive Lebesgue measure, then the coanalytic Toeplitz operator T^*g on the Hardy space H2(\D) is weakly hypercyclic. On the contrary, if g(\D)∩\D=\varnothing, |g|>1 almost everywhere on \T and log(|g|-1)∈ L1(\T), then T^*g is not 1-weakly hypercyclic and hence is not weakly hypercyclic (a bounded linear operator T on a complex Banach space X is called n-weakly hypercyclic if there is x∈ X such that for every surjective continuous linear operator S:X→ \Cn, the set \S(Tmx):m∈\N\ is dense in \Cn). The last result is based upon lower estimates of the norms of the members of orbits of a coanalytic Toeplitz operator. Finally, we show that there is a 1-weakly hypercyclic operator on a Hilbert space, whose square is non-cyclic and prove that a Banach space operator is weakly hypercyclic if and only if it is n-weakly hypercyclic for every n∈\N.