Hankel Operators on Weighted Fock Spaces

Abstract.

We consider Hankel operators $H_{\bar{f}}$ with antiholomorphic symbol $\bar{f}$ on the generalized Fock space ${\mathcal{A}}^{2}(\mu_{m})$, where μ _m is the measure with weight $e^{{-|z|}^{m}}$, m > 0 with respect to the Lebesgue measure in ${\mathbb{C}}^{n}$. We prove that $H_{\bar{f}}$ is bounded if and only if f is a polynomial of degree at most $\frac{m}{2}$. We show that $H_{\bar{f}}$ is compact if and only if f is a polynomial of degree strictly smaller that $\frac{m}{2}$. We also establish that $H_{\bar{f}}$ is in the Schatten class ${\mathcal{S}}_{p}$ if and only if p > 2n and f is a polynomial of degree strictly smaller than $m\frac{(p-2n)}{2p}$.