Integral, differential and multiplication operators on generalized Fock spaces

TL;DR

This paper investigates the boundedness and compactness of Volterra integral and multiplication operators on generalized Fock spaces with fast-decaying weights, revealing nonexistence of nontrivial symbols for certain operators and unboundedness of the differential operator.

Contribution

It extends the analysis of Volterra and multiplication operators to a broad class of generalized Fock spaces with specific weight conditions, including cases with infinite exponents.

Findings

No nontrivial holomorphic symbols induce bounded Volterra and multiplication operators.

Characterization of bounded and compact Volterra-type integral operators between weighted Fock spaces.

The differential operator acts unboundedly on these 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.