Norm estimates of weighted composition operators pertaining to the Hilbert matrix

Lindström, Mikael; Miihkinen, Santeri; Wikman, Niklas

doi:10.1090/proc/14437

Norm estimates of weighted composition operators pertaining to the Hilbert matrix
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by Mikael Lindström, Santeri Miihkinen and Niklas Wikman

Proc. Amer. Math. Soc. 147 (2019), 2425-2435

DOI: https://doi.org/10.1090/proc/14437

Published electronically: March 1, 2019

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Abstract:

Very recently, Božin and Karapetrović [J. Funct. Anal. 274 (2018), no. 2, pp. 525–543] 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 $H^\infty _\alpha$ for $0 < \alpha \le 2/3$ and an upper bound for the norm on the scale $2/3 < \alpha < 1$.

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