Products of Volterra Type Operators and Composition Operators Between Fock Spaces

We show that entire functions $$\varphi $$, which induce bounded products of Volterra integral operators $$V_g$$ (Volterra companion operators $$J_g$$) and composition operators $$C_{\varphi }$$ acting between different Fock spaces, must be affine functions, i.e. $$\varphi (z) = az + b$$. Then, using this special form of $$\varphi $$, we characterize boundedness and compactness of these products in term of new quantities, which are much simpler than the Berezin type integral transforms in the previous papers.