Spectral properties of Volterra-type integral operators on Fock--Sobolev spaces

TL;DR

This paper investigates the spectral properties of Volterra-type integral operators on Fock--Sobolev spaces, establishing conditions for boundedness, compactness, Schatten class membership, and spectrum characterization based on the symbol polynomial.

Contribution

It provides a complete characterization of boundedness, compactness, Schatten class inclusion, and spectral properties of Volterra-type operators on Fock--Sobolev spaces, linking these to polynomial degree and coefficients.

Findings

Boundedness of V_g characterized by degree ≤ 2 polynomial g.

Compactness of V_g characterized by degree ≤ 1 polynomial g.

Spectrum of V_g is a closed disk determined by polynomial coefficients.

Abstract

We study some spectral properties of Volterra-type integral operators $V_g$ and $I_g$ with holomorphic symbol $g$ on the Fock--Sobolev spaces $\mathcal{F}_{\psi_m}^p$. We showed that $V_g$ is bounded on $\mathcal{F}_{\psi_m}^p$ if and only if $g$ is a complex polynomial of degree not exceeding two, while compactness of $V_g$ is described by degree of $g$ being not bigger than one. We also identified all those positive numbers $p$ for which the operator $V_g$ belongs to the Schatten $\mathcal{S}_p$ classes. Finally, we characterize the spectrum of $V_g$ in terms of a closed disk of radius twice the coefficient of the highest degree term in a polynomial expansion of $g$.