Generalized Volterra Integral Operators on Fock Spaces

In this paper, we extend the Voleterra integral operator \(V_g\) and its companion \(J_g\) to integral operator

$$\begin{aligned} T_g^{n,m}f(z)=\int _0^z f^{(n)}(w) g^{(m)}(w)dw. \end{aligned}$$

Using a unified approach, we completely characterize the boundedness and compactness of \(T_g^{n,m}\) from one Fock space \(F_\alpha ^p\) to another \(F_\beta ^q\) for \(0<p,q\le \infty \), \(0<\alpha ,\beta <\infty \). As a surprising case, we obtain that the boundedness (compactness) of \(V_g\) and \(J_g\) from \(F_\alpha ^p\) to \(F_\beta ^q\) is equivalent when the weight parameter \(\alpha <\beta \). We also estimate the norms and essential norms of \(T_g^{n,m}\).