The multiplicative Hilbert matrix
Ole Fredrik Brevig, Karl-Mikael Perfekt, Kristian Seip, Aristomenis G., Siskakis, Dragan Vukoti\'c

TL;DR
This paper introduces the multiplicative Hilbert matrix, analyzes its spectral properties, and compares it to the classical Hilbert matrix, revealing its norm, spectrum, and boundedness characteristics on various function spaces.
Contribution
It defines a new multiplicative Hilbert matrix, establishes its norm and spectral properties, and explores its boundedness on different Dirichlet and Hardy spaces, connecting it to classical results.
Findings
The norm of the multiplicative Hilbert matrix is π.
It has a purely continuous spectrum equal to [0, π].
The matrix is unbounded on certain Hardy spaces for p ≠ 2.
Abstract
It is observed that the infinite matrix with entries for appears as the matrix of the integral operator with respect to the basis ; here is the Riemann zeta function and is defined on the Hilbert space of Dirichlet series vanishing at 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 . 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 and that it has a purely continuous spectrum…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
