Dynamics and spectra of composition operators on the Schwartz space

Abstract

In this paper we study the dynamics of the composition operators defined in the Schwartz space S(R) of rapidly decreasing functions. We prove that such an operator is never supercyclic and, for monotonic symbols, it is power bounded only in trivial cases. For a polynomial symbol ϕ of degree greater than one we show that the operator is mean ergodic if and only if it is power bounded and this is the case when ϕ has even degree and lacks fixed points. We also discuss the spectrum of composition operators.

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

“…Remark 5. 3 We point out that Proposition 5.1 is a consequence of [10,Proposition 4.4]. Since the proof given here does not depend on the open mapping theorem, the result can be generalized to other contexts.…”

“…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).…”

Mean ergodic composition operators on spaces of smooth functions and distributions

“…Suppose that 0 < ψ ′ (x 1 ) < 1 and 0 < ψ ′ (x 2 ) < 1, then there are y 1 , y 2 ∈]x 1 , x 2 [ such that ψ(y 1 ) < y 1 and ψ(y 2 ) > y 2 . Thus by Bolzano's Theorem there is a fixed point between y 1 and y 2 which contradicts (10) and proves the Auxiliary Claim 1.…”

“…In recent years there have been several articles studying mean ergodicity and related properties of (weighted) composition operators on various spaces of functions, such as spaces of holomorphic functions in finite dimensions [8], [5], [4], [12], [3], [16], [23], spaces of holomorphic functions on infinite Banach spaces [18], spaces of homogeneous polynomials on infinite dimensional Banach spaces [17], spaces of real analytic functions [9], the Schwartz space of rapidly decreasing functions on R [10], spaces of meromorphic functions [11], and within the general framework of function spaces defined by local properties [19]. This note is organized as follows.…”

Linear Dynamical Systems

“…In contrast, it was shown in [58] and [72] that the Fock and Schwartz spaces do not even support, respectively, supercyclic weighted and unweighted composition operators.…”