The following argument is contained in Benzi, Boito, and Razouk (2013):
If \(H_{\ell k} = 0\) for all \(r_{\ell k} \geq m\), then
\[\begin{align} [H^n]_{\ell k} = \sum_{\ell_1,\dots,\ell_{n-1}} H_{\ell \ell_1} H_{\ell_1\ell_2} \cdots H_{\ell_{n-1} k} = 0 \end{align}\]
for all \(r_{\ell k} \geq nm\).
Therefore, for all \(n \leq \frac1m r_{\ell k}\), we have
\[\begin{align} \left| (z - H)^{-1}_{\ell k} \right| % &= \inf_{P_n \in \mathcal P_n} \left| \left[ (z - H)^{-1} - P_n(H) \right]_{\ell k} \right| \nonumber\\ % &\leq \inf_{P_n \in \mathcal P_n} \left\| (z - \cdot)^{-1} - P_n \right\|_{L^\infty(\sigma(H))} \nonumber\\ % &\lesssim e^{-\eta \, n} \approx e^{-\frac{\eta}m \, r_{\ell k} } \end{align}\]
where \(\eta \sim \mathrm{dist}\big( z, \sigma(H) \big)\).
Example: Here, we let \(p_n\) be the polynomial interpolation of \(x \mapsto (z - x)^{-1}\) on Chebyshev nodes \(\{ \cos \frac{j\pi}{n} \}_{j=0}^n\) and plot the \(L^\infty\) error on \([-1,1]\) for different choices of \(z\) (this gives an upper bound on the locality of the resolvent as above):
This is \(O(e^{-\eta \, n})\) decay with
┌─────────────────┬───────────┐ │ dist( z, σ(H) ) │ η │ ├─────────────────┼───────────┤ │ 1.0 │ 0.880853 │ │ 0.5 │ 0.479982 │ │ 0.25 │ 0.24404 │ │ 0.125 │ 0.116845 │ │ 0.0625 │ 0.0505203 │ └─────────────────┴───────────┘
i.e. it roughly halves when the distance to the “spectrum” is halved.
References
Benzi, Michele, Paola Boito, and Nader Razouk. 2013. “Decay Properties of Spectral Projectors with Applications to Electronic Structure.” SIAM Rev. 55 (1): 3–64. https://doi.org/10.1137/100814019.