$G$ be a Lie group or algebra of ${\mathcal{M}}(2,{\mathbb{C}})$. We define

-   $\Lambda G$ as the set of smooth functions $f:{\mathbb{S}}^1 \longrightarrow G$.

-   $\Lambda G_\rho\subset \Lambda G$ as the set of functions $f$ such that ${\left\Vertf\right\Vert}_{\rho}$ is finite. If $G$ is a group (or an algebra) then $(\Lambda G_\rho,{\left\Vert\cdot\right\Vert}_{\rho})$ is a Banach Lie group (or algebra).

-   $\Lambda G_\rho^{\geq 0}\subset\Lambda G_\rho$ as the set of functions $f$ such that $f_i=0$ for all $i<0$.

-   $\Lambda_+ G_\rho\subset \Lambda G_\rho^{\geq 0}$ as the set of functions such that $f_0$ is upper-triangular.

-   $\Lambda_+^{\mathbb{R}}{\mathrm{SL}}(2,{\mathbb{C}})_\rho\subset \Lambda_+ {\mathrm{SL}}(2,{\mathbb{C}})_\rho$ as the set of functions that have positive elements on the diagonal.

We also define $\Lambda{\mathbb{C}}$ as the set of smooth maps from ${\mathbb{S}}^1$ to ${\mathbb{C}}$, and $\Lambda{\mathbb{C}}_\rho$ and $\Lambda {\mathbb{C}}_\rho^{\geq 0}$ as above. Note that every function of $\Lambda G_\rho$ holomorphically extends to ${\mathbb{A}}_\rho$ and that every function of $\Lambda G_\rho^{\geq 0}$ holomorphically extends to ${\mathbb{D}}_\rho$.

We will use the Fröbenius norm on ${\mathcal{M}}(2,{\mathbb{C}})$: $$\left| A \right| := \left(\sum_{i,j} |a_{ij}|^2\right)^{\frac{1}{2}}.$$ Recall that this norm is sub-multiplicative. Therefore, the norm ${\left\Vert\cdot\right\Vert}_{\rho}$ is sub-multiplicative. Moreover, $$\forall A \in {\Lambda {\mathrm{SL}}(2,{\mathbb{C}})_\rho}, \quad {\left\VertA^{-1}\right\Vert}_\rho = {\left\VertA\right\Vert}_\rho,$$ $$\forall A\in\Lambda{\mathcal{M}}(2,{\mathbb{C}})_\rho,\quad \forall {\lambda}\in{\mathbb{A}}_{\rho},\quad \left|A({\lambda})\right|\leq{\left\VertA\right\Vert}_{\rho}.$$

The DPW method relies on the Iwasawa decomposition. 