Spectrums and Uniform Mean Ergodicity of Weighted Composition Operators on Fock Spaces

For holomorphic pairs of symbols $$(u, \psi )$$ ( u , ψ ) , we study various structures of the weighted composition operator $$ W_{(u,\psi )} f= u \cdot f(\psi )$$ W ( u , ψ ) f = u · f ( ψ ) defined on the Fock spaces $$\mathcal {F}_p$$ F p . We have identified operators $$W_{(u,\psi )}$$ W ( u , ψ ) that have power-bounded and uniformly mean ergodic properties on the spaces. These properties are described in terms of easy to apply conditions relying on the values |u(0)| and $$|u(\frac{b}{1-a})|$$ | u ( b 1 - a ) | , where a and b are coefficients from linear expansion of the symbol $$\psi $$ ψ . The spectrum of the operators is also determined and applied further to prove results about uniform mean ergodicity.