The multiplicative Hilbert matrix

Abstract

It is observed that the infinite matrix with entries $(\sqrt{mn}\log (mn){)}^{-1}$ for $m,n\ge 2$ appears as the matrix of the integral operator $\mathbf{H}f(s):={\int }_{1/2}^{+\infty }f(w)(\zeta (w+s)-1)dw$ with respect to the basis $(n^{-s}{)}_{n\ge 2}$; here $\zeta (s)$ is the Riemann zeta function and $H$ is defined on the Hilbert space ${\mathcal{H}}_0^2$ of Dirichlet series vanishing at $+\infty$ and with square-summable coefficients. This infinite matrix defines a multiplicative Hankel operator according to Helson's terminology or, alternatively, it can be viewed as a bona fide (small) Hankel operator on the infinite-dimensional torus ${\mathbb{T}}^{\infty }$. By analogy with the standard integral representation of the classical Hilbert matrix, this matrix is referred to as the multiplicative Hilbert matrix. It is shown that its norm equals $\pi$ and that it has a purely continuous spectrum which is the interval $[0,\pi ]$; these results are in agreement with known facts about the classical Hilbert matrix. It is shown that the matrix $(m^{1/p}{n}^{(p-1)/p}\log (mn){)}^{-1}$ has norm $\pi /\sin (\pi /p)$ when acting on ${\ell }^p$ for $1<p<\infty$. However, the multiplicative Hilbert matrix fails to define a bounded operator on ${\mathcal{H}}_0^p$ for $p\ne 2$, where ${\mathcal{H}}_0^p$ are $H^p$ spaces of Dirichlet series. It remains an interesting problem to decide whether the analytic symbol ${\sum }_{n\ge 2}(\log n{)}^{-1}{n}^{-s-1/2}$ of the multiplicative Hilbert matrix arises as the Riesz projection of a bounded function on the infinite-dimensional torus ${\mathbb{T}}^{\infty }$.

A number of modifications made based on two referee reports

References in corpus (1)

• Failure of Nehari's Theorem for Multiplicative Hankel Forms in Schatten Classes

Cited by in corpus (10)

• Linear space properties of $H^p$ spaces of Dirichlet series
• Riesz projection and bounded mean oscillation for Dirichlet series
• Contractive inequalities for Bergman spaces and multiplicative Hankel forms
• On Helson matrices: moment problems, non-negativity, boundedness, and finite rank
• Weak product spaces of Dirichlet series
• Volterra operators on Hardy spaces of Dirichlet series
• Composition operators and embedding theorems for some function spaces of Dirichlet series
• The critical point and the $p$-norm of $A_s$ and $C$-matrices
• The spectrum of some Hardy kernel matrices
• Volterra operators and Hankel forms on Bergman spaces of Dirichlet series