Operators on l∞ with totally disconnected spectrum and applications to J-class operators

In this note we show that the spectrum of the adjoint of any operator on l∞ consists entirely of eigenvalues under the assumption that the spectrum is totally disconnected. In particular we will show that J-class operators on l∞ have at least uncountable spectrum. The notion of J-sets and therewith the J-class operators on Banach spaces were introduced by A. Manoussos and G. Costakis and certain classes like weighted backward shifts were studied under this new concept of dynamic behavior. This type of operators can be seen as localized topological transitive operators which are in particular interesting in the case where the Banach space is l∞ since it was shown by T. Bermúdez and N.J. Kalton that no topological transitive operator can exist on l∞. Under some additional conditions we show that the point spectrum has even an interior point using some techniques from tauberian operator theory.