Year: 2026
Type: preprint
Abstract: For $α>1$, we study the weighted multiplicative Hilbert operator \[ (T_αf)(s) = \int_{1/2}^{\infty} f(t)\bigl(ζ(s+t)^α-1\bigr) \left(t-\frac12\right)^{α-1}\,dt \] on the Hilbertian Bergman space $A_{α,0}^2$ of Dirichlet series with zero constant term. In the natural orthonormal basis, $T_α$ is represented by the weighted multiplicative Hilbert matrix \[ \left( \frac{Γ(α)\sqrt{d_α(m)d_α(n)}} {\sqrt{mn}\,(\log(mn))^α} \right)_{m,n\ge2}, \] where $d_α$ is the generalized divisor function defined by $ζ(s)^α=\sum_{n\ge1}d_α(n)n^{-s}$ for $\Re s>1$. Set $Λ_α=Γ(α/2)^2/Γ(α)$. We prove that the essential spectrum of \(T_α\) is \([0,Λ_α]\) and that the spectrum in this interval is purely absolutely continuous with multiplicity one. Above \(Λ_α\), there are at most finitely many eigenvalues, all of which are simple. Moreover, there is a unique \(α_*\in(1,2)\) such that \(T_α\) has eigenvalues if and only if \(α>α_*\). Consequently, the associated sharp embedding inequality admits a non-zero extremizer precisely when $α>α_*$, and this extremizer is unique up to scalar multiplication. We also establish the asymptotic formula $\|T_α\|\simΓ(α+1)/(2(2\log 2)^α)$ as $α\to\infty$. A key step in the proof combines Mellin-transform asymptotics, strict total positivity, and eigenvalue counting to exclude embedded eigenvalues. The proof includes a general result on the essential and singular continuous spectra of weighted integral Hankel operators.
Source: arXiv (Cornell University)
Authors: Ma, Tengfei, Lu, Yufeng, Zu, Chao
Institution: Dalian University of Technology
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204166778