Integral, Differential and Multiplication Operators on Generalized Fock Spaces

Abstract

Volterra companion integral and multiplication operators with holomorphic symbols are studied for a large class of generalized Fock spaces on the complex plane $\CC$. The weights defining these spaces are radial and subject to a mild smoothness condition. In addition, we assumed that the weights decay faster than the classical Gaussian weight. One of our main results show that there exists no nontrivial holomorphic symbols $g$ which induce bounded Volterra companion integral $I_g$ and multiplication operators $M_g$ acting between the weighted spaces. We also describe the bounded and compact Volterra-type integral operators $V_g$ acting between $\mathcal{F}_q^\psi$ and $\mathcal{F}_p^\psi$ when at least one of the exponents $p$ or $q$ is infinite, and extend results of Constantin and Pel\'{a}ez for finite exponent cases. Furthermore, we showed that the differential operator $D$ acts in unbounded fashion on these and the classical Fock spaces.