Recurrence of multiples of composition operators on weighted Dirichlet spaces. (English) Zbl 1495.47023
Summary: A bounded linear operator \(T\) acting on a Hilbert space \(\mathcal{H}\) is said to be recurrent if for every non-empty open subset \(U\subset \mathcal{H}\) there is an integer \(n\) such that \(T^n (U)\cap U\neq \emptyset\). In this paper, we completely characterize the recurrence of scalar multiples of composition operators, induced by linear fractional self maps of the unit disk, acting on weighted Dirichlet spaces \(\mathcal{S}_\nu\); in particular on the Bergman space, the Hardy space, and the Dirichlet space. Consequently, we complete previous work of Costakis, Manoussos, and Parissis [G. Costakis et al., Complex Anal. Oper. Theory 8, No. 8, 1601–1643 (2014; Zbl 1325.47019)] on the recurrence of linear fractional composition operators on Hardy space. In this manner, we determine the triples \((\lambda,\nu,\phi)\in \mathbb{C}\times \mathbb{R}\times \mathrm{LFM}(\mathbb{D})\) for which the scalar multiple of composition operator \(\lambda C_\phi\) acting on \(\mathcal{S}_\nu\) fails to be recurrent.
MSC:
| 47A16 | Cyclic vectors, hypercyclic and chaotic operators |
| 47B33 | Linear composition operators |
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
| 46E50 | Spaces of differentiable or holomorphic functions on infinite-dimensional spaces |
| 46T25 | Holomorphic maps in nonlinear functional analysis |
Keywords:
hypercyclicity; recurrence; composition operator; Dirichlet spaces; Bergman space; Hardy spaceCitations:
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