The fine spectra of a difference of weighted shift operators acting in some classical sequence spaces

In this paper we investigate the fine spectra, the ergodic properties and the linear dynamics of the operatorC(x)=(−nxn+n(n−1)xn−2)n∈N0,x=(xn)n∈N0∈CN0, and of its formal adjoint C′ acting in the classical sequence spaces s, s′, ω and φ. The operators C and C′ acting in s are conjugate to the one-dimensional Ornstein-Uhlenbeck differential operator Ou=u″−xu′ and to the differential operator O′u=u″+xu′+u acting in S(R), respectively.