A note on the differential operator on generalized Fock spaces

TL;DR

This paper investigates the spectral properties of the differential operator on generalized Fock spaces, identifying conditions for compactness, Schatten class membership, and characterizing its spectrum as the closed unit disk.

Contribution

It characterizes the spectral structures of the differential operator on various Fock type spaces, including compactness and Schatten class membership, and determines its spectrum.

Findings

The operator admits spectral structures like compactness and Schatten class membership.

Its spectrum on these spaces is exactly the closed unit disk.

Provides new insights into the spectral theory of differential operators on generalized Fock spaces.

Abstract

It has long been known that the differential operator $D$ represents a typical examples of unbounded operators in many Banach spaces including the classical Fock spaces, the Fock--Sobolev spaces, and the generalized Fock spaces where the weight decays faster than the Gaussian weight. In this note we identify Fock type spaces where the operator admits some basic spectral structures including compactness and membership in the Schatten $\mathcal{S}_p$ classes. We also showed that its nontrivial spectrum while acting on such spaces is precisely the closed unit disk $\D$ in the complex plane.