Generalized Volterra Integral Operators on Fock Spaces

In this paper, we extend the Voleterra integral operator \(V_g\) $V_g$ and its companion \(J_g\) $J_g$ to integral operator \(\begin{aligned} T_g^{n,m}f(z)=\int _0^z f^{(n)}(w) g^{(m)}(w)dw. \end{aligned}\) $$\begin{array}{c}\hfill T_g^{n,m}f(z)={\int }_0^z{f}^{(n)}(w)g^{(m)}(w)dw.\end{array}$$ Using a unified approach, we completely characterize the boundedness and compactness of \(T_g^{n,m}\) $T_g^{n,m}$ from one Fock space \(F_\alpha ^p\) $F_{\alpha }^p$ to another \(F_\beta ^q\) $F_{\beta }^q$ for \(0<p,q\le \infty \) $0<p,q\le \infty$ , \(0<\alpha ,\beta <\infty \) $0<\alpha ,\beta <\infty$ . As a surprising case, we obtain that the boundedness (compactness) of \(V_g\) $V_g$ and \(J_g\) $J_g$ from \(F_\alpha ^p\) $F_{\alpha }^p$ to \(F_\beta ^q\) $F_{\beta }^q$ is equivalent when the weight parameter \(\alpha <\beta \) $\alpha <\beta$ . We also estimate the norms and essential norms of \(T_g^{n,m}\) $T_g^{n,m}$ .