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

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 $g$.

Cited by in corpus (1)

• On the spectrum of Volterra-type integral operators on Fock--Sobolev spaces