For the Segal-Bargmann space $H^2(\mathbb{C}^n, \mu)$ of Gaussian square integrable entire functions on $\mathbb{C}^n$ we consider Hankel operators $H_f$ with $f$ in a symbol space $\mathcal{T}(\mathbb{C}^n)$. We define the Berezin transform for operators on $H^2(\mathbb{C}^n,\mu)$ and in terms of the {\itshape mean oscillation} of $f$ we give necessary and sufficient conditions for $H_f$ and $H_{\bar{f}}$ to be bounded, compact or to belong to the {\itshape von Neumann Schatten class} $\mathcal{S}_p$ for $1\leq p <\infty$. We compare some aspects of these results to the case of Bergman spaces over bounded symmetric domains.

There are close relations to the boundedness of the commutator $[P,M_f]$ where $P$ denotes the Toeplitz projection and $M_f$ is the multiplication by $f$. We describe how to construct spectral invariant Fréchet operator algebras $(\Psi_k^{\Delta})_k$ with prescribed properties in $\mathcal{L}(H^2(\mathbb{C}^n, \mu))$ similar to the Hörmander classes $\Psi_{\rho, \delta}^0$ of zero-order pseudo-differential operators. Following general ideas by B. Gramsch we are using commutator methods and finite systems $\Delta$ of closed operators. With $\Delta$ in a class of vector fields on $\mathbb{C}^n$ this construction leads to algebras localized in cones $\mathcal{C}\subset \mathbb{C}^n$ and containing the Segal-Bargmann projection.

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