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Norm estimates of weighted composition operators pertaining to the Hilbert matrix


Authors: Mikael Lindström, Santeri Miihkinen and Niklas Wikman
Journal: Proc. Amer. Math. Soc. 147 (2019), 2425-2435
MSC (2010): Primary 47B38; Secondary 30H20
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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Additional Information

Mikael Lindström
Affiliation: Department of Mathematics, Åbo Akademi University, FI-20500 Åbo, Finland
Email: mikael.lindstrom@abo.fi

Santeri Miihkinen
Affiliation: Department of Mathematics, Åbo Akademi University, FI-20500 Åbo, Finland
Email: santeri.miihkinen@abo.fi

Niklas Wikman
Affiliation: Department of Mathematics, Åbo Akademi University, FI-20500 Åbo, Finland
Email: niklas.wikman@abo.fi

DOI: https://doi.org/10.1090/proc/14437
Keywords: Hilbert matrix, weighted composition operator, operator norm, Bergman spaces, Korenblum spaces
Received by editor(s): May 20, 2018
Published electronically: March 1, 2019
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2019 American Mathematical Society