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

Abstract

In this paper we investigate the spectra and the ergodic properties of the multiplication operators and the convolution operators acting on the Schwartz space $${\mathcal S}({\mathbb R})$$ S ( R ) of rapidly decreasing functions, i.e., operators of the form $$M_h: {\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ M h : S ( R ) → S ( R ) , $$f \mapsto h f $$ f ↦ h f , and $$C_T:{\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ C T : S ( R ) → S ( R ) , $$f\mapsto T\star f$$ f ↦ T ⋆ f . Precisely, we determine their spectra and characterize when those operators are power bounded and mean ergodic.

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

“…For further results on mean ergodic operators we refer to [19,21]. For recent results on mean ergodic operators in lcHs' we refer to [1,2,3,4,5,6], for example, and the references therein.…”

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

“…For further results on mean ergodic operators we refer to [19,22]. For recent results on mean ergodic operators in lcHs' we refer to [1][2][3][4]7,12,17], for example, and the references therein.…”

On diagonal operators between the sequence (LF)-spaces $$l_p({\mathcal {V}})$$

“…The case of multiplication operators on weighted spaces of analytic functions on the complex unit disc was studied by Bonet and Ricker [11]. Albanese and the author in [5] have studied the spectra and the ergodic properties of the multiplication operators on the space S(R) of rapidly decreasing functions and the (PLB)-space of its multipliers. Echelon and co-echelon spaces were studied by Köthe and Toeplitz.…”