On the Spectra of Multiplicative Hilbert Operator on Bergman Spaces of Dirichlet Series
Abstract
For α>1, we study the weighted multiplicative Hilbert operator (Tαf)(s) = ∫1/2∞ f(t)(ζ(s+t)α−1) (t−1/2)α−1 dt on the Hilbertian Bergman space Aα,02 of Dirichlet series with zero constant term. In the natural orthonormal basis, Tα is represented by the weighted multiplicative Hilbert matrix ( Γ(α)√(dα(m)dα(n))/(√(mn) (log(mn))α) )m,n≥2, where dα is the generalized divisor function defined by ζ(s)α=∑n≥1dα(n)n−s for ℜ s>1. Set Λα=Γ(α/2)2/Γ(α). We prove that the essential spectrum of Tα is [0,Λα] and that the spectrum in this interval is purely absolutely continuous with multiplicity one. Above Λα, there are at most finitely many eigenvalues, all of which are simple. Moreover, there is a unique α*∈(1,2) such that Tα 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 |Tα|∼Γ(α+1)/(2(2log 2)α) as α→∞. 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.
Related
- Norm of the generalized Hilbert operator on weighted Bergman spaces
- The Hilbert matrix operator into the Bergman space
- The Differential Hilbert Operator Between Weighted Bergman Spaces
- The Bergman projection with operator weights
- Sharp weighted norm estimates for the positive Bergman operator: the two-parameter case
BibTeX
@misc{ma2026spectra,
title = {{On the Spectra of Multiplicative Hilbert Operator on Bergman Spaces of Dirichlet Series}},
author = {Tengfei Ma and Yufeng Lu and Chao Zu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.22882},
eprint = {2608.22882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22882},
}
RIS
TY - UNPB
TI - On the Spectra of Multiplicative Hilbert Operator on Bergman Spaces of Dirichlet Series
AU - Tengfei Ma
AU - Yufeng Lu
AU - Chao Zu
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2608.22882
UR - https://arxiv.org/abs/2608.22882
AB - For α>1, we study the weighted multiplicative Hilbert operator (T_αf)(s) = ∫_1/2^∞ f(t)(ζ(s+t)^α−1) (t−1/2)^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 ( Γ(α)√(d_α(m)d_α(n))/(√(mn) (log(mn))^α) )_m,n≥2, where d_α is the generalized divisor function defined by ζ(s)^α=∑_n≥1d_α(n)n^−s for ℜ 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_*∈(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|∼Γ(α+1)/(2(2log 2)^α) as α→∞. 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.
ER -