The norm of a Finite Hilbert matrix

Question

Let $H$ be an $n\times n$ Hilbert matrix, $$h_{ij}=(i+j-1)^{-1}.$$

The matrix $p$-norm corresponding to the p-norm for vectors is: $\left \| A \right \| _p = \sup \limits _{x \ne 0} \frac{\left \| A x\right \| _p}{\left \| x\right \| _p}$, $p\ge 1$.

Is there a known (or what is the) formula for $\left \| H \right \| _p$ in terms of $n$ and $p$?

I saw a related post for $n=\infty$. https://math.stackexchange.com/questions/837832/norm-of-hilbert-matrix-is-it-equal-to-pi

After reading the post Spectral norm for a truncated Hilbert matrix I guess the answer to my query is no... so a big challenge.

Answer 1

I think the answer is no, indeed. Even in the particular case $p=2$ the formula is not known (to my best knowledge). Concerning $\|H\|_{2}$, is known $\|H\|_{2}\leq\pi$. More precisely, we have the following upper bound $$\|H\|_{2}\leq 2w_n\arcsin\frac{1}{w_n}$$ where $$w_n=2\left[\binom{2n}{n}\right]^{-1/2n},$$ see P. Otte: Upper bounds for the spectral radius of the $n\times n$ Hilbert matrix, Pacific J. Math. 219 No. 2 (2005) 101-109.

In addition, quite precise asymptotic formula for $\|H\|_{2}$ is known as $n\to\infty$. Namely, $$\|H\|_{2}=\pi-\frac{\pi^{5}}{2\ln^{2} n}+O\left(\frac{\ln\ln n}{\ln^{3} n}\right),$$ see N.G. de Bruijn and H.S. Wilf, On Hilbert’s inequality in n dimensions, Bull. Amer. Math. Soc. 68 (1962), 70–73.