Positive Toeplitz operators on pluriharmonic Fock space: Schatten class criteria and sharp norm comparisons

Let $\mu$ be a positive Borel measure on $\C^n$. For every $0<p<\infty$, we prove that the Toeplitz operator $T_\mu^{\mathrm{ph}}$ induced by $\mu$ on pluriharmonic Fock space belongs to $\Sp_p$ if and only if $z\mapsto\mu(B(z,r))$ belongs to $L^p(\C^n)$ for one, or equivalently every, $r>0$; this is also equivalent to Schatten membership of the corresponding holomorphic Toeplitz operator $T_\mu$. For $n\geq2$, this settles a conjecture of Jaguzovi\'c and Vujadinovi\'c, and the result includes the range $0<p<1$ in every dimension. Moreover, $$\norm{T_\mu}_{\Sp_p}^p \leq\norm{T_\mu^{\mathrm{ph}}}_{\Sp_p}^p \leq2^{\max\{1,p\}}\norm{T_\mu}_{\Sp_p}^p,$$ and both constants are optimal. More generally, we obtain a sharp comparison for every symmetrically normed ideal. The proof uses the holomorphic and antiholomorphic splitting: positivity controls the mixed block by the diagonal blocks, while a square root factorization of the positive block operator yields the singular value estimates. We also obtain an exact trace identity.