Convex-Cyclic Weighted Composition Operators and Their Adjoints

Abstract

We characterize the convex-cyclic weighted composition operators $$W_{(u,\psi )}$$ W ( u , ψ ) and their adjoints on the Fock space in terms of the derivative powers of $$ \psi $$ 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].

Integral and Weighted Composition Operators on Fock-type Spaces

“…As illustrated in the diagram above, pointwise topology is weaker than the weak and strong topologies on F p ϕ and hence the space supports no supercyclic (weakly) weighted composition operators. The analogous statement on the classical Fock spaces was proved in [7]. It turns out that the same conclusion remains enforce for F p ϕ and hence the unboundedness of the Laplacian has no effect in this regard.…”

“…It follows the map ψ fixes the point z 0 = b 1−a for a = 1 and z 0 = 0, or a = 1 and b = 0. Then, the rest of the proof follows exactly the same arguments used in the proof of Theorem 1.7 in [7].…”

“…The various linear dynamical structures of W (u,ψ) on the classical Fock spaces are now well-understood including convex-cyclicity [7]. Whether the unboundedness of the Laplacian of ϕ has an effect on the dynamical structures as it does for boundedness, compactness and Schatten class membership is another noteworthy problem to investigate.…”