Comparing positive Toeplitz operators on holomorphic and pluriharmonic Fock spaces

Abstract

We prove sharp norm comparisons between positive Toeplitz operators associated with the same positive Borel measure, acting respectively on holomorphic functions and on sums of holomorphic and antiholomorphic functions, both equipped with the Gaussian-weighted $L^2$ norm. For every positive Borel measure $\mu$ on ${\mathbb{C}}^n$ and $0<p<\infty$, both operators are compact with summable $p$th powers of their singular values (i.e. Schatten) exactly when $z\mapsto \mu (B(z,r))$ belongs to $L^p({\mathbb{C}}^n)$ for some fixed radius $r>0$. For other weights, this agreement can fail dramatically. We construct a weight $e^{-2\varphi }$ and a finite positive measure $\mu$ on $\mathbb{C}$ for which the operator on holomorphic functions has the same singular values as with the Gaussian weight and the same measure $\mu$. These values have summable $p$th powers for every $p>0$, yet the quadratic form on sums of holomorphic and antiholomorphic functions is unbounded in weighted $L^2$. The example still satisfies $\Delta \varphi =2$, as for the Gaussian choice $\varphi (z)=|z{|}^2/2$.

arXiv comment25 pages. v3: Streamlined exposition, with greater emphasis on the example where the holomorphic operator belongs to every Schatten class while the pluriharmonic Toeplitz form is unbounded