theory is used to give a short proof of an upper bound on the optimal angle of the bounded functional calculus result for the Ornstein-Uhlenbeck operator associated with $\phi$ of the form $\phi(x) = xN(x) + lx$, where $N:\mathbb{R}^d \to \mathbb{R}^d$ is a positive definite real-symmetric linear operator, and $l \in \mathbb{R}^d$. This result returns the known optimal angle for the classical Ornstein-Uhlenbeck operator, for which $N=\frac{1}{2}I, l=0$. The proof depends on knowing the Weyl symbol for the semigroup generated by the Ornstein-Uhlenbeck operator, which we provide. This explicit knowledge of the symbol in the classical case is also crucial to the results of the paper [@vNPWeylGaussian].

The space $HS_0(M)$ is then examined further, with an aim to find symbols for the semigroup generated by the relevant Ornstein-Uhlenbeck operator outside of the quadratic case. We show that $HS_0(M)$ can be expanded and endowed with a certain product to form the unital Banach algebra $(HS(M), \#)$, with $HS(M) \subset L^\infty\left(\mathbb{R}^{2d}\right)$ and $\#$ the Moyal product, in which case the generalised Weyl calculus can be extended in a natural way such that it becomes a contractive Banach algebra homomorphism $HS(M) \to B(L^p(\mu))$.

Some ideas are then presented, about how the symbol class 