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Diagonal (multiplication) operators acting between a particular class of countable inductive spectra of Fréchet sequence spaces, called sequence (LF)-spaces, are investigated. We prove results concerning boundedness, compactness, power boundedness, and mean ergodicity. Furthermore, we determine when a diagonal operator is Montel and reflexive. We analyze the spectra in terms of the system of weights defining the spaces.

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.

The aim of this paper is to introduce and to study the space $${{\mathcal {O}}}_{M,\omega }({{\mathbb {R}}}^N)$$ O M , ω ( R N ) of the multipliers of the space $${{\mathcal {S}}}_\omega ({{\mathbb {R}}}^N)$$ S ω ( R N ) of the $$\omega $$ ω -ultradifferentiable rapidly decreasing functions of Beurling type. We determine various properties of the space $${{\mathcal {O}}}_{M,\omega }({{\mathbb {R}}}^N)$$ O M , ω ( R N ) . Moreover, we define and compare some lc-topologies of which $${{\mathcal {O}}}_{M,\omega }({{\mathbb {R}}}^N)$$ O M , ω ( R N ) can be naturally endowed.