The Differential Hilbert Operator Between Weighted Bergman Spaces

In this paper, a complete characterization of the boundedness, compactness, norm and essential norm of the differential Hilbert operator ${\mathcal{H}}_2:A_{\alpha }^2\to A_{\beta }^2$ is obtained. More precisely, ${\mathcal{H}}_2:A_{\alpha }^2\to A_{\beta }^2$ is bounded if and only if $-1<\alpha <0$ and $\beta \ge \alpha +2$ and it is compact if and only if $-1<\alpha <0$ and $\beta >\alpha +2$. The norm and essential norm of $\mathrm{\parallel }{\mathcal{H}}_2{\mathrm{\parallel }}_{A_{\alpha }^2\to A_{\beta }^2}$ are also investigated. In particular, when $\beta =\alpha +2$, $$\mathrm{\parallel }{\mathcal{H}}_2{\mathrm{\parallel }}_{A_{\alpha }^2\to A_{\alpha +2}^2}=\mathrm{\parallel }{\mathcal{H}}_2{\mathrm{\parallel }}_{\mathrm{e},A_{\alpha }^2\to A_{\alpha +2}^2}=\frac{\pi \sqrt{(\alpha +2)(\alpha +3)}}{\sin (\pi (\alpha +2)\mathrm{/}2)},-1<\alpha <0.$$ When $\beta >\alpha +2$, $\mathrm{\parallel }{\mathcal{H}}_2{\mathrm{\parallel }}_{\mathrm{e},A_{\alpha }^2\to A_{\beta }^2}=0$. Furthermore, for every $1\le p<\mathrm{\infty }$, the operator ${\mathcal{H}}_2$ belongs to the Schatten class ${\mathcal{S}}_p$ if and only if it is compact.