Journal of the
Korean Mathematical Society JKMS

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 $g$.

Keywords: Fock space, Fock--Sobolov spaces, bounded, compact, Volterra integral, multiplication operators, Schatten class, spectrum, generalized Fock spaces

MSC numbers: Primary 47B32, 30H20; Secondary 46E22, 46E20, 47B33