The norm of the Hilbert matrix operator on Bergman spaces

Key Points

• The operator norm of the Hilbert matrix on the Bergman space equals \(\frac{\pi}{\sin((2+\alpha)\pi/p)}\) within a specific range of \(\alpha\), demonstrating a key mathematical property.
• Evidence supports this conjecture for \(0 \leq \alpha \leq \frac {6p^3-29p^2+17p-2+2p\sqrt {6p^2-11p+4}}{(3p-1)^2}\), refining the previously accepted limits.
• The study employs theoretical analysis to prove the validity of the conjecture, utilizing methods from functional analysis to explore operator behavior in Hilbert spaces.
• This finding enhances understanding of operator norms and provides a sharper boundary for applications in complex analysis and mathematical operator theory.