Generalized composition operators on weighted Fock spaces

The generalized composition operators $J_g^{\varPhi}$ and $C_g^{\varPhi}$, induced by analytic functions $g$ and $\varPhi$ on the complex plane $\mathbb{C}$, are defined by
\begin{align*} J_g^{\varPhi}(f)(z)=\int\limits_0^zf'(\varPhi(\omega))g(\omega)d\omega \ \text{and} \ C_g^{\varPhi}(f)(z)=\int\limits_0^{\varPhi(z)}f'(\omega)g(\omega)d\omega. \end{align*}
In this paper, we consider these operators on weighted Fock spaces $\mathcal{F}_p^{\varPsi}$, consisting of entire functions, which are $\mathcal{L}^p(\mathbb{C})$-integrable with respect to the measure $d\lambda(z)=e^{-\varPsi (z)}d\Lambda(z)$, where $d\Lambda$ is the usual Lebesgue area measure in $\mathbb{C}$. We assume that the weight function $\varPsi$ in the spaces satisfies certain smoothness conditions, in particular, this weight function grows faster than the Gaussian weight $\frac{|z|^2}{2}$ defining the classical Fock spaces. We first consider bounded and compact properties of $J_{g}^ {\varPhi}$ and $C_{g}^{\varPhi}$, and characterize these properties in terms function theory of inducing functions $g$ and $\varPhi$, given by
\begin{align*} \mathcal{M}_g^{\varPhi}(z):=\frac{|g(z)|\varPsi'(\varPhi(z))}{1+\varPsi'(z)}e^{\varPsi(\varPhi(z))-\varPsi(z)}. \end{align*}
Our characterization is simpler to use than the Berezin type integral transform characterization. In some cases, our result shows that these operators experience poorer boundedness and compactness structures when acting between such spaces than the classical Fock spaces. For instance, for $\varPhi(z)=z$, there is no nontrivial bounded $J_g^{\varPhi}$ and $C_g^{\varPhi}$ on weighted Fock spaces. In the case of classical Fock spaces, they are bounded if and only if $g$ is constant. In the second part of this paper, we apply our simpler characterization of boundedness and compactness to further study the Schatten-class membership of these operators. In particular, we express the Schatten $S_p(\mathcal{F}_2^{\varPsi})$ class membership property in terms of $\mathcal{L}^p(\mathbb{C}, \Delta \varPsi d\Lambda)$-integrability of $\mathcal{M}_g^{\varPhi}$.