Dynamical properties of weighted translation operators on the Schwartz space $$\mathcal {S}(\mathbb {R})$$S(R)

Abstract

In this paper we investigate the dynamical properties of weighted translation operators acting on the Schwartz space S(R) of rapidly decreasing functions, i.e., operators of the form T w : S(R) → S(R), f (•) → w(•) f (•+1). We characterize when those operators are hypercyclic, weakly mixing, mixing and chaotic. Several examples illustrate our results and show which of those classes are different.

Spectra and ergodic properties of multiplication and convolution operators on the space $${\mathcal S}({\mathbb R})$$

“…In [8] the authors characterized the multipliers h ∈ O M (R) such that M h : S(R) → S(R) has closed range. We also mention that in the last years the study of the properties, like closed range and dynamical behaviour, of the composition operators acting on the Schwartz space S(R) has been considered by several authors (see [10][11][12][13] for examples and the references therein).…”

Characterization of hypercyclic weighted composition operators on the space of holomorphic functions

“…Weighted composition operators play an important role in functional analysis. Their dynamical properties were intensively studied over the past decades: see [4,7,13,24] for composition operators on spaces of holomorphic functions, [5,16,22,25] for weighted composition operators on spaces of holomorphic functions, [8] for composition operators on spaces of analytic functions, [20] for weighted composition operators on spaces of smooth functions, [15] for composition operators on spaces of functions defined by local properties, see [12] for weighted translation operators acting on the Schwartz space.…”