A note on composition operators between weighted spaces of smooth functions

Abstract

For certain weighted locally convex spaces and of one real variable smooth functions, we characterize the smooth functions : R → R for which the composition operator : → , ↦ → • is well-defined and continuous. This problem has been recently considered for = being the space S of rapidly decreasing smooth functions [1] and the space O of slowly increasing smooth functions [2]. In particular, we recover both these results as well as obtain a characterization for = being the space O of very slowly increasing smooth functions.

On composition operators between weighted (LF)‐ and (PLB)‐spaces of continuous functions

“…Recently, some authors have considered the problem to characterize the well-posedness and the continuity of composition operators in the setting of (PLB)-spaces or (LF)-spaces of smooth functions on ℝ 𝑁 , like the space  𝑀 (ℝ 𝑁 ) of the slowly increasing smooth functions and the space  𝐶 (ℝ 𝑁 ) of the very slowly increasing smooth functions, see [2,10]. While, in [8] it has been characterized when continuous multiplication operators on a weighted inductive limit of Banach spaces of continuous functions are power bounded, mean ergodic or uniformly mean ergodic.…”