The Differential Hilbert Operator Between Weighted Bergman Spaces

Abstract

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 $\Vert {\mathcal{H}}_2{\Vert }_{A_{\alpha }^2\to A_{\beta }^2}$ are also investigated. In particular, when $\beta =\alpha +2$, \[ \|\mathcal H_2\|_{A^2_α\to A^2_{α+2}} =\|\mathcal H_2\|_{\mathrm e,A^2_α\to A^2_{α+2}} =\frac{π\sqrt{(α+2)(α+3)}} {\sin(π(α+2)/2)},\qquad -1<α<0. \] When $\beta >\alpha +2$, $\Vert {\mathcal{H}}_2{\Vert }_{\mathrm{e},A_{\alpha }^2\to A_{\beta }^2}=0$. Furthermore, for every $1\le p<\infty$, the operator ${\mathcal{H}}_2$ belongs to the Schatten class ${\mathcal{S}}_p$ if and only if it is compact.