{\mathbb{R}}$; maybe, we lose analyticity but rough estimates – when just the absolute values of matrix elements work well – are good enough.

Indeed, , could be rewritten as $$\begin{gathered}
R^0 = K^2, \quad U_j = K (K W K)^j K, \label{eq:ujrevise}\\
R = R^0 + \sum_{j = 1}^{\infty} K (K W K)^j K,\end{gathered}$$ where $$(K W K)_{k m} = \frac{1}{\sqrt{z - z_k}} W_{km} \frac{1}{\sqrt{z - z_m}},\, \, k, m = 0, 1, 2, \dotsc, \, \, z \in {\mathbb{C}}\setminus {\mathbb{R}}. \label{eq:kwkform}$$

\[lem:hs\] Under the assumptions , and , with ${\displaystyle}0 < \alpha < \frac{1}{2}$, if $z \in H_n \setminus \mathcal{D}_n$, then $KWK$ is a Hilbert-Schmidt operator, and $$\ell {\equiv}\Vert K W K \Vert_{\text{HS}} \leq \frac{C_0 M(\alpha) \log (en)}{n^{2\alpha}} \label{eq:HSnorm}, \quad M(\alpha) {\equiv}6 + \frac{4/3}{1- 2 \alpha} + \frac{1}{3\alpha}$$

If $z \in {\partial (\mathcal{D}_n )}$, i.e, $$\begin{aligned}
\begin{aligned}
z = (2n + 1) + \xi + i \eta , & & &\quad \vert \xi \vert = \frac{1}{2}, \, \vert \eta \vert \leq \frac{1}{2},\\
&& \text{ or } & \quad \vert \xi \vert \leq \frac{1}{2}, \, \vert \eta \vert = \frac{1}{2}; \quad \xi, \eta \in {\mathbb{R}},
\end{aligned} \label{eq:bdloc}\end{aligned}$$ then $$\frac{1}{2} \leq \vert z - z_j \vert \leq 3, \, j = 