~k=0, \, 1, \, 2, \, 3$ that maps functions from $\CySpT^{m, \,
    k}$ onto the functional spaces defined in Cartesian coordinates $$\begin{aligned}
\eta_{m, \, 0}&: \CySpT^{m, \, 0}  \rightarrow \CaSpB^0, & u & \mapsto
\dfrac{m}{\rho} u, \\ 
\eta_{m, \, 1}&: \CySpT^{m, \, 1}  \rightarrow \CaSpB^1 \times
\CaSpB^0, & 
\uv & \mapsto 
\begin{bmatrix}
\dfrac{1}{\rho} \parT{m\,u_\rho + u_\theta}\\
\dfrac{m}{\rho}u_z\\
u_\theta
\end{bmatrix},\\
\eta_{m, \, 2} &: \CySpT^{m, \, 2}  \rightarrow \CaSpB^{1*} \times
\CaSpB^2, & 
\uv & \mapsto
\begin{bmatrix}
u_\rho\\
u_z\\
\dfrac{1}{\rho}\parT{mu_\theta - u_\rho}
\end{bmatrix}, \\
\eta_{m, \, 3} &: \CySpT^{m, \, 3} \rightarrow \CaSpB^{2}, &u &
\mapsto u~. \end{aligned}$$ These operators are chosen such that $\vec{G}$, $\vec{C}$ and $D$ are both well-defined and make the top part of the diagram  commutative. Note that, if $\Gamma^0 \neq \emptyset$, these operators are well-defined only on regular subspaces $\CySpTT^{m, \, k}\subset\CySpT^{m,
    \, k}$ where $$\label{eq:Zm_tilde}
\begin{aligned}
\CySpTT^{m, \, 0} & = \parG{u \in \CySpT^{m, \, 0}:~ \eta_{m, \, 0}(u)
    \in \CaSpB^0}~, \\ 
\CySpTT^{m, \, 1} & = \parG{\uv \in \CySpT^{m, \, 1}:~ \eta_{m, \,
        1}(\uv) \in \CaSpB^1 \times \CaSpB^0}~, \\ 
\CySpTT^{m, \, 2} & = \parG{\uv \in \CySpT^{m, \, 2}:~ \eta_{m, \,
        2}(\uv) \in \CaSpB^{1*} \times \CaSpB^2}~, \\ 
\CySpTT^{m, \, 3} & = \parG{u \in 