发表机构
California Institute of Technology（加州理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对多重调和Fock空间上的正Toeplitz算子，建立了其Schatten类准则，解决了相关猜想，得到了与对应全纯Toeplitz算子的精确范数比较，还获得了对称范数理想的精确比较及迹恒等式。
AI 中文摘要
设μ为复空间ℂⁿ上的正Borel测度，对任意0<p<∞，我们证明：多重调和Fock空间上由μ诱导的Toeplitz算子T_μ^ph属于Schatten类Sp_p，当且仅当映射z↦μ(B(z,r))属于L^p(ℂⁿ)，且该条件对任意一个r>0成立等价于对所有r>0均成立；这也等价于对应的全纯Toeplitz算子T_μ的Schatten类性质。当n≥2时，该结果解决了Jaguzović与Vujadinović的一个猜想，且在所有维数下均包含0<p<1的范围。此外，我们得到范数比较式：||T_μ||_{Sp_p}^p ≤ ||T_μ^ph||_{Sp_p}^p ≤ 2^{max{1,p}}||T_μ||_{Sp_p}^p，且两个常数均为最优。更一般地，我们对每个对称范数理想均得到了精确比较。证明采用全纯与反全纯分解：正性通过对角块控制混合块，而正块算子的平方根分解可得到奇异值估计。我们还得到了一个精确的迹恒等式。
英文摘要
Let $μ$ be a positive Borel measure on $\mathbb C^n$. We prove that, for every $0<p<\infty$, the Toeplitz operator $T_μ^{\mathrm{ph}}$ induced by $μ$ on the pluriharmonic Fock space belongs to the Schatten class $S_p$ if and only if the local mass function $z\mapstoμ(B(z,r))$ belongs to $L^p(\mathbb C^n)$ for one, or equivalently every, $r>0$. For $n\geq2$, this resolves a conjecture of Jaguzovi'c and Vujadinovi'c, and the range $0<p<1$ is new in every dimension. Writing $T_μ$ for the corresponding holomorphic Toeplitz operator, we obtain the sharp estimates $$\|T_μ\|_{S_p}^p \leq \|T_μ^{\mathrm{ph}}\|_{S_p}^p \leq 2^{\max\{1,p\}} \|T_μ\|_{S_p}^p. $$ We also prove a sharp comparison with constant two in every symmetrically normed ideal and an exact trace formula, using positivity and a $2\times2$ block decomposition whose diagonal blocks are $T_μ$ and an antiunitary copy of its compression to the functions orthogonal to constants. We then consider generalized Fock weights satisfying $m \, dd^c|z|^2\le dd^cϕ\le M \, dd^c|z|^2$. For the canonical holomorphic and antiholomorphic direct sum norm, the same Schatten and symmetrically normed ideal estimates hold. For the norm inherited from $L^2(\mathbb C^n,e^{-2ϕ}dV)$, the local mass criterion also holds whenever $e^{-2ϕ}$ is comparable to a generalized Fock weight invariant under the scalar circle action. Without further assumptions, the local mass criterion can fail for the inherited norm: in one complex dimension, we construct a weight of the form $ϕ(z)=|z|^2/2+\operatorname{Re}q(z)$, with $q$ entire, and a finite positive measure whose local masses belong to every $L^p$, although the Toeplitz form on the inherited pluriharmonic space is unbounded.
Comments35 pages. Comments welcome. v2: Added results for generalized Fock weights and the inherited norm; expanded the discussion
