The specification property for composition operators

Abstract

It is known that the operator specification property does not coincide, in general, with other notions of chaos in linear dynamics. In this paper, we show that, for a natural and widely studied class of linear operators, namely the dissipative composition operators of bounded distortion (including the class of weighted shifts), specification-type properties do not introduce new dynamical behaviors: they coincide with classical notions of chaos. We also extend these equivalences with conditions on periodic points and examine in detail the special case of the translation operator $T_α$ acting on $L^p(\mathbb R,\mathcal B,μ)$, where the measure $μ$ is induced by a density function. Moreover, we show that, in general, there is no relationship between the operator specification property and the shadowing property in the linear context. Finally, we analyse the important role played by the bounded distortion hypothesis.