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 operator C (x) = ( - nx(n) + root n(n - 1 )x(n-2))(n is an element of N0) , x = (x(n))(n is an element of N0 )is an element of C-N0 , and of its formal adjoint C ' acting in the classical sequence spaces s , s ' , omega and phi . The operators C and C ' acting in s are conjugate to the one-dimensional OrnsteinUhlenbeck differential operator Ou = u '' - xu ' and to the differential operator O ' u = u '' + xu ' + u acting in S ( R), respectively.