Helson Forms and Integral Operators on Bergman Spaces of Dirichlet Series

Abstract

We study a family of operators $T_α$, $α>1$, on Hilbertian Bergman spaces of Dirichlet series. In the natural orthonormal basis, these operators are represented by weighted multiplicative Hilbert matrices involving the generalized divisor coefficients $d_α$. We determine their spectral type. The essential and absolutely continuous spectra are $[0,Λ_α]$, with absolutely continuous multiplicity one; the singular continuous spectrum is empty, and there are no embedded eigenvalues. There are only finitely many eigenvalues above $Λ_α$, and all of them are simple. We also prove that there is a unique $α_*\in(1,2)$ such that no eigenvalues occur above $Λ_α$ for $1<α<α_*$, whereas such eigenvalues exist for every $α>α_*$. As part of the proof, we establish a spectral theorem for a general class of weighted integral Hankel operators with kernels $w(x)b(x+y)\overline{w(y)}$. Finally, we study the corresponding weighted Helson forms, give sufficient conditions for boundedness and compactness, and characterize boundedness for forms induced by finite positive measures.