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

Abstract

In this paper we consider composition operators on locally convex spaces of functions defined on R. We prove results concerning supercyclicity, power boundedness, mean ergodicity and convergence of the iterates in the strong operator topology.

A note on composition operators between weighted spaces of smooth functions

“…

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.

…”

“…By setting ( ) = ( ) = 1 + | | in Theorem 1 we recover the above results about S and O from [1,2] as well as the following characterization for the space O of very slowly increasing smooth functions:…”

“…Albanese et. al [2] proved that the composition operator : O → O , with O the space of slowly increasing smooth functions [3], is well-defined (continuous) if and only if ∈ O . In [2,Remark 2.6] they also pointed out that the corresponding result for the space O of very slowly increasing smooth functions [3] is false, namely, they showed that sin( 2 ) ∉ O , while, obviously, sin , 2 ∈ O .…”