$HS(M)$ and its properties as a Banach algebra of functions could be used to study Ornstein-Uhlenbeck like operators, such as determining symbols for relevant semigroups and a factorisation of the functional calculus through the space $HS(M)$.

Initial Definitions {#secInitialDefinitions}
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The work of van Neerven and Portal in [@vNPWeylPairs] develops a Weyl calculus for $2d$-tuples of operators satisfying the following definition:

\[defnWeylPair\] A $2d$-tuple $(Q,P) = (Q_1, \ldots, Q_d, P_1, \ldots, P_d)$ of closed, densely defined operators on a Banach space is called a Weyl pair if each operator generates a uniformly bounded group $\exp(ix_iQ_i), \exp(i\xi_iP_i)$ which satisfy the integrated canonical commutation relations $$\begin{array}{c}
\exp(ix_iQ_i)\exp(ix_jQ_j) = \exp(ix_jQ_j)\exp(ix_iQ_i), \forall i,j = 1,... d\\
\exp(i\xi_iP_i)\exp(i\xi_jP_j) = \exp(i\xi_jP_j)\exp(i\xi_iP_i), \forall i,j = 1,... d\\
\exp(ixQ)\exp(i\xi P) = \exp(ix\xi)\exp(i\xi P)\exp(ixQ).\\
\end{array}$$ In which case we define $$\exp(i(xQ + \xi P)) = \exp(\frac{1}{2}ix\xi)\exp(ixQ)\exp(i\xi P), \forall x,\xi \in \mathbb{R}^d.$$

This definition of $\exp(i(xQ + \xi P))$ for Weyl pairs can be motivated by the Baker-Campbell-Hausdorff formula, for example, noting that formally differentiating the integrated commutation relations produces the formal commutation relation “$[Q_i, P_j] = i \delta_{ij}I$". It is easy to check that this definition does make the set of operators $\{\exp(i(xQ + \xi P)); x, \xi \in 