Abstract
We study the cyclic structures of the weighted composition operators and their adjoints on the Fock space {mathcal {F}}_2. A complete characterization of cyclicity which depends on the derivative of the symbol for the composition operator and non-vanishing structure of the weight function is provided. It is further shown that the space fails to support supercyclic adjoint weighted composition operators. As a tool in proving our main results, we also identified eigenvectors of the weighted composition operators in the space which is interest of its own.
Highlights
For a pair of entire functions (u, ψ) on the complex plane C, the induced weighted composition operator W(u,ψ) maps f to u f (ψ)
We study the dynamical properties of the operators and their adjoints on the Fock space F2
In [13], we reported that there exists no supercyclic composition operator on Fock spaces
Summary
For a pair of entire functions (u, ψ) on the complex plane C, the induced weighted composition operator W(u,ψ) maps f to u f (ψ). If u = 1, W(u,ψ) is just the composition map Cψ : f → f (ψ). If ψ is the identity map, W(u,ψ) reduces to the multiplication operator Mu : f → u f. W(u,ψ) generalizes the two operators and can be written as a product W(u,ψ) = MuCψ. The theory of weighted composition operators traces back to the sixties in the work of Forelli [6] where it was shown that the isometries on the Hardy spaces H p whenever 1 < p < ∞
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