Supercyclicity of weighted composition operators on spaces of continuous functions

• Autores: María José Beltrán Meneu, Enrique Jordá, Marina Murillo Arcila
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 71, Fasc. 3, 2020, págs. 493-509
• Idioma: inglés
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• Resumen

  • Our study is focused on the dynamics of weighted composition operators defined on a locally convex space E\hookrightarrow (C(X),\tau _p) with X being a topological Hausdorff space containing at least two different points and such that the evaluations \{\delta _x:\ x\in 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,\varphi } is never weakly supercyclic on E. We also prove that if the symbol \varphi lies in the unit ball of A(\mathbb {D}), then every weighted composition operator can never be \tau _p-supercyclic neither on C(\mathbb {D}) nor on the disc algebra A(\mathbb {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.