狄利克雷级数哈代空间上有限素数复合算子的临界范数轮廓
Critical Norm Profiles for Finite-Prime Composition Operators on the Hardy Space of Dirichlet Series
浏览论文内容
中文总结 AI 辅助
本文研究狄利克雷级数哈代空间上有限素数复合算子，确定其临界边界范数轮廓，推导相关算子的收敛关系、结构性质、集中不等式及有限维近似，揭示单一系数算子结构的支配作用。
中文摘要 AI 辅助
我们确定狄利克雷级数的哈代-希尔伯特空间\\(\mathcal H^2\\)上有限素数复合算子的临界边界算子范数轮廓。对于\\(\varphi_{\delta,\boldsymbol\rho}(s) = \frac12+\delta + \delta\sum_{j=1}^d\rho_jp_j^{-s}\\)（其中\\(\boldsymbol\rho\in B_d\\)），重正化正系数算子在算子范数下以\\(O(\delta)\\)的误差一致收敛到显式多元加权汉克尔算子\\(\mathcal H_{\boldsymbol\rho}\\)；因此\\(2\delta\\|C_{\varphi_{\delta,\boldsymbol\rho}}\\|^2 = \\|\mathcal H_{\boldsymbol\rho}\\| + O(\delta)\\)在\\(B_d\\)上一致成立。我们证明极限算子具有全次数约化\\(\mathcal H_{\boldsymbol\rho} \simeq D_{\boldsymbol\rho} H_{R_{\boldsymbol\rho}/2} D_{\boldsymbol\rho}\oplus\mathbf{0}\\)，其中对角因子是归一化素数权重的卷积碰撞范数。该结构结合Brevig和Perfekt的仿射比较原理，得到\\(\\|\mathcal H_{\boldsymbol\rho}\\|\\)的显式集中不等式，确定单素数构型是极限范数估计中的精确等号情形，并给出远离该情形的定量亏缺。对于固定的\\(\sigma>\frac12\\)，我们还得到范数平方的二阶展开式以及具有显式全次数和狄利克雷和截断误差的全有限维近似。这些结果共同表明，单一系数算子结构支配奇异边界轮廓、固定\\(\sigma\\)的微扰 regime 以及已验证的有限维近似。
英文摘要
We identify the critical boundary operator-norm profile of finite-prime composition operators on the Hardy--Hilbert space \(\mathcal H^2\) of Dirichlet series. For \[ φ_{δ,\boldsymbolρ}(s) = \frac12+δ+ δ\sum_{j=1}^dρ_jp_j^{-s}, \qquad \boldsymbolρ\in B_d, \] the renormalized positive coefficient operators converge uniformly in operator norm, with \(O(δ)\) error, to an explicit multivariate weighted Hankel operator \(\mathcal H_{\boldsymbolρ}\); consequently, \[ 2δ\|C_{φ_{δ,\boldsymbolρ}}\|^2 = \|\mathcal H_{\boldsymbolρ}\| + O(δ) \] uniformly over \(B_d\). We show that the limiting operator admits the total-degree reduction \[ \mathcal H_{\boldsymbolρ} \simeq D_{\boldsymbolρ} H_{R_{\boldsymbolρ}/2} D_{\boldsymbolρ}\oplus\mathbf{0}, \] where the diagonal factors are convolution-collision norms of the normalized prime weights. This structure, together with the affine comparison principle of Brevig and Perfekt, yields an explicit concentration inequality for \(\|\mathcal H_{\boldsymbolρ}\|\), identifies the one-prime configurations as the exact equality cases in the limiting norm estimate, and gives a quantitative deficit away from them. For fixed \(σ>\frac12\), we also obtain a second-order expansion of the squared norm and fully finite-dimensional approximations with explicit total-degree and Dirichlet-sum truncation errors. Together, these results show that a single coefficient-operator structure governs the singular boundary profile, the fixed-\(σ\) perturbative regime, and certified finite-dimensional approximation.
