Helson Forms and Integral Operators on Bergman Spaces of Dirichlet Series

Abstract

We study a family of operators $T_\alpha$, $\alpha>1$, 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 $d_\alpha$. We determine their spectral type. The essential and absolutely continuous spectra are $[0,\Lambda_\alpha]$, with absolutely continuous multiplicity one; the singular continuous spectrum is empty, and there are no embedded eigenvalues. There are only finitely many eigenvalues above $\Lambda_\alpha$, and all of them are simple. We also prove that there is a unique $\alpha_*\in(1,2)$ such that no eigenvalues occur above $\Lambda_\alpha$ for $1<\alpha<\alpha_*$, whereas such eigenvalues exist for every $\alpha>\alpha_*$. 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.