Convex-cyclic weighted composition operators and their adjoints

We characterize the convex-cyclic weighted composition operators W(u,ψ) and their adjoints on the Fock space in terms of the derivative powers of ψ and the location of the eigenvalues of the operators on the complex plane. Such a description is also equivalent to identifying the operators or their adjoints for which their invariant closed convex sets are all invariant subspaces. We further show that the space supports no supercyclic weighted composition operators with respect to the pointwise convergence topology and, hence, with the weak and strong topologies, and answers a question raised by T. Carrol and C. Gilmore in [5]