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

Let $\mu$ be a positive Borel measure on ${\text{C}}^n$. For every $0<p<\mathrm{\infty }$, we prove that the Toeplitz operator $T_{\mu }^{\mathrm{p}\mathrm{h}}$ induced by $\mu$ on pluriharmonic Fock space belongs to ${\text{Sp}}_p$ if and only if $z\mapsto \mu (B(z,r))$ belongs to $L^p({\text{C}}^n)$ for one, or equivalently every, $r>0$; this is also equivalent to Schatten membership of the corresponding holomorphic Toeplitz operator $T_{\mu }$. For $n\ge 2$, this settles a conjecture of Jaguzović and Vujadinović, and the result includes the range $0<p<1$ in every dimension. Moreover, $$\text{norm}{T_{\mu }}_{{\text{Sp}}_p}^p\le \text{norm}{T_{\mu }^{\mathrm{p}\mathrm{h}}}_{{\text{Sp}}_p}^p\le 2^{\max \{1,p\}}\text{norm}{T_{\mu }}_{{\text{Sp}}_p}^p,$$ 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.