AI 中文总结
该研究确定了狄利克雷级数伯格曼空间上算子族$T_\alpha$的谱型，证明了$\alpha_*$的存在性，建立了加权积分汉克尔算子的谱定理，并刻画了加权赫尔森型的有界性等性质。
AI 中文摘要
我们研究希尔伯特型狄利克雷级数伯格曼空间上的算子族$T_\alpha$（$\alpha>1$）。在自然正交基下，这些算子由含广义除数系数$d_\alpha$的加权乘法希尔伯特矩阵表示。我们确定了它们的谱型：本质谱和绝对连续谱为$[0,\Lambda_\alpha]$，绝对连续重数为1；奇异连续谱为空，无嵌入本征值；仅存在有限个大于$\Lambda_\alpha$的本征值，且均为单本征值。我们还证明存在唯一的$\alpha_*\in(1,2)$，使得当$1<\alpha<\alpha_*$时，$\Lambda_\alpha$上方无本征值，而当$\alpha>\alpha_*$时存在此类本征值。作为证明的一部分，我们建立了一类核为$w(x)b(x+y)\overline{w(y)}$的加权积分汉克尔算子的谱定理。最后，我们研究了相应的加权赫尔森型，给出了有界性与紧性的充分条件，并刻画了有限正测度诱导型的有界性。
英文摘要
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.
Comments79 pages
