Supercyclicity of weighted composition operators on spaces of continuous functions

Abstract

Our study is focused on the dynamics of weighted composition operators defined on a locally convex space E ֒→ (C(X), τ p ) with X being a topological Hausdorff space containing at least two different points and such that the evaluations {δ x : x ∈ X} are linearly independent in E ′ . We prove, when X is compact and E is a Banach space containing a nowhere vanishing function, that a weighted composition operator C w,ϕ is never weakly supercyclic on E. We also prove that if the symbol ϕ lies in the unit ball of A(D), then every weighted composition operator can never be τ psupercyclic neither on C(D) nor on the disc algebra A(D). Finally, we obtain Ansari-Bourdon type results and conditions on the spectrum for arbitrary weakly supercyclic operators, and we provide necessary conditions for a composition operator to be weakly supercyclic on the space of holomorphic functions defined in non necessarily simply connected planar domains. As a consequence, we show that no composition operator can be weakly supercyclic neither on the space of holomorphic functions on the punctured disc nor in the punctured plane.

Convex-Cyclic Weighted Composition Operators and Their Adjoints

“…Observe also that f cannot have zero in C. if not, all the elements in the projective orbit will also vanish at a possible zero which extends to the closure and contradicts. Thus, by Proposition 4 of [3], for any two different numbers z, w ∈ C, Then, we arrive at the desired conclusion by simply replacing the compact set K by Q b in the above argument and completes the proof.…”

Dynamics of composition operators on function spaces defined by local and global properties

“…Suppose ϕ to be increasing. If ϕ(x) < x for all x > 0, then ϕ n (x) → 0 as n → ∞ for all x > 0 and so C ϕ is not supercyclic by [8,Theorem 8 (ii)]. Hence, ϕ(x) > x for every x > 0.…”

Convex-cyclic weighted composition operators and their adjoints

“…Observe also that f can not have zero in C. if not, all the elements in the projective orbit will also vanish at a possible zero which extends to the closure and contradicts. Thus, by Proposition 4 of [3], for any two different numbers z, w ∈ C, Then we arrive at the desired conclusion by simply replacing the compact set K by Q b in the above argument and completes the proof.…”