Norm estimates of weighted composition operators pertaining to the Hilbert Matrix

Abstract

Very recently, Božin and Karapetrović solved a conjecture by proving that the norm of the Hilbert matrix operator $\mathcal{H}$ on the Bergman space $A^p$ is equal to $\frac{\pi }{\sin (\frac{2\pi }{p})}$ for $2<p<4.$ In this article we present a partly new and simplified proof of this result. Moreover, we calculate the exact value of the norm of $\mathcal{H}$ defined on the Korenblum spaces $H_{\alpha }^{\infty }$ for $0<\alpha \le 2/3$ and an upper bound for the norm on the scale $2/3<\alpha <1$.

10 pages