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 ${\alpha }_{\ast }\in (1,2)$ such that no eigenvalues occur above ${\Lambda }_{\alpha }$ for $1<\alpha <{\alpha }_{\ast }$, whereas such eigenvalues exist for every $\alpha >{\alpha }_{\ast }$. 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.

79 pages