SlowGuess/NeurIPS_2023_OCR huggingface dataset:
While we keep assumptions (BK) and (SD) as in [18, 29], assumption (RC) is, up to our knowledge, novel. The rationale behind it is that for  $\alpha = 1$  (RC) is equivalent to  $\operatorname{Im}(A_{\pi}S) \subseteq \operatorname{Im}(S)$ , in which case there exists a bounded  $\pi$ -a.e. Koopman operator  $G_{\mathcal{H}} \colon \mathcal{H} \to \mathcal{H}$  [24]. On the other hand, as  $\alpha \to 0$  (RC) becomes closer to  $\operatorname{Im}(A_{\pi}S) \subseteq \operatorname{cl}(\operatorname{Im}(S))$  which is always satisfied for universal kernels since  $\operatorname{cl}(\operatorname{Im}(S)) = L_{\pi}^{2}(\mathcal{X})$  [38, Chapter 4]. Importantly, as the next example shows, (RC) is weaker condition than the usual regularity conditions; see Appendix D.1 for a detailed discussion.
Example 2. Let  $X$  be an  $\mathcal{X}$ -valued random variable with law  $\pi$ . Consider the Markov chain  $(X_{t})_{t\in \mathbb{N}}$  such that  $X_{t} = X$  for all  $t\in \mathbb{N}$ . Then  $\pi$  is an invariant measure and  $A_{\pi} = I_{L_{\pi}^{2}(\mathcal{X})}$  is the identity map on  $L_{\pi}^{2}(\mathcal{X})$ . Clearly, (RC) holds for all  $\alpha \in (0,1]$ . On the other hand, since  $A_{\pi}S = SG_{\mathcal{H}}$  for bounded operator  $G_{\mathcal{H}} = I_{\mathcal{H}}\not\in\mathrm{HS}(\mathcal{H})$ , HS-norm learning rates derived in [29] do not apply.
In order to study the error of any empirical finite rank estimator  $\widehat{G}$  we rely on the error decomposition
$$
\mathcal {E} (\widehat {G}) \leq \underbrace {\| A _ {\pi} S - S G _ {\gamma} \|} _ {\text {r e g u l a r i z a t i o n b i a s}} + \underbrace {\| S \left(G _ {\gamma} - G\right) \|} _ {\text {r a n k r e d u c t i o n b i a s}} + \underbrace {\| S (G - \widehat {G}) \|} _ {\text {e s t i m a t o r s v a r i a n c e}}, \tag {14}
$$
where  $G_{\gamma} \coloneqq C_{\gamma}^{-1}T$  is the minimizer of the full (i.e. without rank constraint), Tikhonov regularized, HS norm error, and  $G$  is the population version of the empirical estimator  $\widehat{G}$ .
While the last two terms in the r.h.s. of (14) depend of the estimator of choice, the first term depends only on the choice of  $\mathcal{H}$  and the regularity of  $A_{\pi}$  w.r.t.  $\mathcal{H}$ . In this work we focus on the classical kernel-based learning of the Koopman operator [20, 24, 29], where one chooses a universal kernel [38] Chapter 4] for which  $\operatorname{Im}(A_{\pi}S) \subseteq \operatorname{cl}(\operatorname{Im}(S))$ , and controls the regularization bias with a regularity condition. For details see Rem. [7] of Appendix D.2
The second source of bias and the estimator's variance in our error decomposition depends on the choice of the low rank estimator. While throughout this section we consider  $(\mathbf{RC})$  for  $\alpha \in [1,2]$ , we discuss extensions of our results to  $\alpha < 1$  in Appendix D.5
Theorem 2. Assume the operator  $A_{\pi}$  satisfies  $\sigma_r(A_{\pi}S) > \sigma_{r + 1}(A_{\pi}S)\geq 0$  for some  $r\in \mathbb{N}$ . Let (SD) and (RC) hold for some  $\beta \in (0,1]$  and  $\alpha \in [1,2]$ , respectively, and let  $\operatorname {cl}(\operatorname {Im}(S)) = L_{\pi}^{2}(\mathcal{X})$ . Let
$$
\gamma \asymp n ^ {- \frac {1}{\alpha + \beta}} \text {a n d} \varepsilon_ {n} ^ {\star} := n ^ {- \frac {\alpha}{2 (\alpha + \beta)}}. \tag {15}
$$
Let  $\delta \in (0,1)$ . Then, there exists a constant  $c > 0$ , depending only on  $\mathcal{H}$ , such that for large enough  $n \geq r$ , with probability at least  $1 - \delta$  in the i.i.d. draw of  $\mathcal{D}_n$  from  $\rho$
$$
\mathcal {E} (\widehat {G}) \leq \left\{ \begin{array}{l l} \sigma_ {r + 1} \left(A _ {\pi} S\right) + c \varepsilon_ {n} ^ {\star} \ln \delta^ {- 1} & \text {i f} \widehat {G} = \widehat {G} _ {r, \gamma} ^ {\text {R R R}}, \\ \sigma_ {r + 1} (S) + c \varepsilon_ {n} ^ {\star} \ln \delta^ {- 1} & \text {i f} \widehat {G} = \widehat {G} _ {r, \gamma} ^ {\text {P C R}} \text {a n d} \sigma_ {r} (S) > \sigma_ {r + 1} (S). \end{array} \right. \tag {16b}
$$
Proof Sketch. The regularization bias is bounded by  $a\gamma^{\frac{\alpha}{2}}$  by Proposition 5 of Appendix D.2. For the RRR estimator, the rank reduction bias is upper bounded by  $\sigma_{r+1}(A_{\pi}S)$ , while for PCR by  $\sigma_{r+1}(S)$ . The bounds on the variance terms critically rely on the well-known perturbation result for spectral projectors reported in Proposition 3, Appendix A. This result is then chained to two versions of the Bernstein inequality in separable Hilbert spaces. The first one is Pinelis-Sakhanenko's inequality and the second is Minsker's inequality extended to self-adjoint HS-operators, Props. 9 and 11 in Appendix D.3.1 respectively. These inequalities provide high probability bounds for the norms
of  $C_{\gamma}^{-1/2}(\widehat{C} - C)$  and  $C_{\gamma}^{-1/2}(\widehat{C} - C)C_{\gamma}^{-1/2}$ , as well as  $C_{\gamma}^{-1/2}(\widehat{T} - T)$  and  $C_{\gamma}^{-1/2}(\widehat{T} - T)C_{\gamma}^{-1/2}$ . Combining the bias due to regularization and variance terms, for both estimators we obtain the balancing equation  $\gamma^{\frac{\alpha}{2}} = \gamma^{-\frac{\beta}{2}}n^{-\frac{1}{2}}\ln \delta^{-1}$ , which yields the optimal choice of  $\gamma$  and the rates.
We stress that the number of samples in the previous theorem depends on the problem's complexity, expressed in the constants  $c_{\mathrm{RRR}} = \frac{1}{\sigma_r^2(A_\pi S) - \sigma_{r + 1}^2(A_\pi S)}$ , and  $c_{\mathrm{PCR}} = \frac{1}{\sigma_r(S) - \sigma_{r + 1}(S)}$ . Namely, the better the separation of singular values, the smaller number of needed samples. Furthermore, analyzing the bounds (16a) and (16b), we see that faster spectral decay is, in general, preferable. For example, for the Gaussian kernel  $\beta$  can be chosen arbitrarily small, yielding the rate  $n^{-1/2}$ . On the other hand, kernels with slow spectral decay for which  $\beta = 1$  can give slower rates between  $n^{-1/4}$  and  $n^{-1/3}$ . Finally, from the variance bounds for RRR and PCR, c.f. Appendix D.3, one can specify constants. Namely, in the slower regime when  $\varepsilon_n^\star > n^{-1/2}$  we have that  $c = a + 7.2\log(10)\sqrt{2c_{\mathcal{H}}} (1 + ac_{\mathcal{H}}^{(\alpha - 1)/2})(\sqrt{c_{\mathcal{H}}}\wedge \frac{b^{\beta / 2}}{\sqrt{1 - \beta}})$ , while in fastest regime  $\varepsilon_n^\star = n^{-1/2}$ , there is a significant difference between RRR and PCR since  $c$  should be multiplied with the constants  $c_{\mathrm{RRR}}$  and  $c_{\mathrm{PCR}}$ , respectively.
As argued in Section 3, the bound (16a) indicates that for rank  $r$  Koopman operators the error converges to zero w.r.t. the number of training samples, while the bias of PCR is strictly positive. Hence, in order to ensure small error for PCR, high values of the rank parameter might be necessary. To theoretically explain this effect, in Theorem 6 of Appendix D.4 we give also lower bounds of operator norm error for the RRR and PCR estimators showing that  $\mathcal{E}(\widehat{G}_{r,\gamma}^{\mathrm{RRR}})$  always concentrates around  $\sigma_{r+1}(A_{\pi}S)$ , while the concentration of  $\mathcal{E}(\widehat{G}_{r,\gamma}^{\mathrm{PCR}})$  around  $\sigma_{r+1}(S)$  depends on the irreducible risk of the learning problem. To illustrate the tightness of the error concentration bounds we present Example 3 (see also Appendix D.4).
Example 3. Let  $\mathcal{X} = \mathbb{R}$ . Consider the 1D equidistant sampling of the Ornstein-Uhlenbeck process, obtained by integrating the Langevin equation of Example  $\boxed{I}$  with  $\beta = 1$  and  $U(x) = x^{2}/2$ , given by  $X_{t} = e^{-1}X_{t-1} + \sqrt{1 - e^{-2}}\epsilon_{t}$ , where  $\{\epsilon_t\}_{t\geq 1}$  are i.i.d. standard Gaussians. For this process it is well-known [34] that  $\pi$  is  $\mathcal{N}(0,1)$  and that  $A_{\pi}$  admits a spectral decomposition  $(\mu_i,f_i)_{i\in \mathbb{N}}$  in terms of Hermite polynomials. We study the family of kernel functions  $k_{\Pi,\nu}(x,x') := \sum_{i\in \mathbb{N}}\mu_{\Pi(i)}^{2\nu}f_i(x)f_i(x')$ , where  $\Pi$  is a permutation of the indices of the eigenvalues and  $\nu$  is a scaling factor. The rationale behind this class of kernels is that by varying  $\Pi$  and  $\nu$  one morphs the original metric structure of  $A_{\pi}$  in a way which is harder and harder to revert when learning from finite sets of data. In particular, for any target rank  $r$ , setting  $\nu := 1/r^2$  and  $\Pi$  to the permutation such that  $i\mapsto 2r - i + 1 (i\leq r)$ ,  $i\mapsto i - r (r + 1\leq i\leq 2r)$  and  $i\mapsto i$  elsewhere, elementary algebra and our concentration bounds give
$$
| \mathcal {E} (\widehat {G} _ {r, \gamma} ^ {\mathrm {P C R}}) - e ^ {- 1 / r} | \lesssim n ^ {- 1 / 2} \ln \delta^ {- 1}, \qquad | \mathcal {E} (\widehat {G} _ {r, \gamma} ^ {\mathrm {R R R}}) - e ^ {- r} | \lesssim n ^ {- 1 / 2} \ln \delta^ {- 1}.
$$
We refer the reader to Figure 7 and to Section 7 for a numerical implementation of this example.
We conclude this section with remarks on the tightness of our statistical analysis of operator norm error. Since discussed results are not the main focus of the paper, we present them in Appendix D.5
Remark 1 (Lower bound). The rate  $\varepsilon_{n}^{\star} = n^{-\frac{\alpha}{2(\alpha + \beta)}}$  guaranteed by (16a) matches the minimax lower bound for the operator norm error when learning finite rank  $A_{\pi}$ . Formal statement and its proof is given in Theorem 7 of Appendix D.5
Remark 2 (Extension to misspecified setting). The optimal rates for HS-norm error of the KRR estimator are developed in [29] under a stronger condition than (RC) In Theorem 9 of Appendix D.6 we extended this analysis to PCR and RRR estimators, deriving the optimal operator norm rates that also cover cases when Koopman operator cannot be properly defined as bounded operator on the chosen RKHS space  $\mathcal{H}$ .
6 Spectral Learning Rates
Collecting all the previous results, we are now ready to present our spectral learning rates for the two estimators in a general form. For brevity, we focus on two different type of bounds in which (i) we analyse the uniform bound for the whole estimated spectra, and (ii) we express the estimators' bias in empirical form to provide an insight into spuriousness of eigenvalues. Moreover, we present only
eigenvalue estimation bounds, noting that the eigenfunction estimation bounds readily follow from [8]. The complete results are presented in detail in Appendix E.
Theorem 3. Let  $A_{\pi}$  be a compact self-adjoint operator. Under the assumptions of Theorem 2 there exists a constant  $c > 0$ , depending only on  $\mathcal{H}$ , such that for every  $\delta \in (0,1)$ , for every large enough  $n \geq r$  and every  $i \in [r]$  with probability at least  $1 - \delta$  in the i.i.d. draw of  $\mathcal{D}_n$  from  $\rho$
$$
\left| \widehat {\lambda} _ {i} - \mu_ {j (i)} \right| \leq \left\{ \begin{array}{l l} \frac {2 \sigma_ {r + 1} \left(A _ {\pi} S\right)}{\sigma_ {r} \left(A _ {\pi} S\right)} + c \varepsilon_ {n} ^ {\star} \ln \delta^ {- 1} & \text {i f} \quad \widehat {G} = \widehat {G} _ {r, \gamma} ^ {\mathrm {R R R}}, \\ \frac {2 \sigma_ {r + 1} (S)}{\left[ \sigma_ {r} \left(A _ {\pi} S\right) - \sigma_ {r + 1} ^ {\alpha} (S) \right] _ {+}} + c \varepsilon_ {n} ^ {\star} \ln \delta^ {- 1} & \text {i f} \quad \widehat {G} = \widehat {G} _ {r, \gamma} ^ {\mathrm {P C R}}. \end{array} \right. \tag {17}
$$
The uniform eigenvalue learning rates for RRR and PCR estimators, differ in the estimator's bias. While the PCR bias has a factor  $\sigma_{r+1}(S)$  in the numerator, RRR has  $\sigma_{r+1}(A_{\pi}S) \leq \sigma_{r+1}(S)$ . The striking difference happens when the Koopman operator is of finite rank. Then, assuming that  $r$  is properly chosen, RRR estimator has no bias, and  $c_{\mathrm{RRR}}$  is typically moderate. On the other hand, PCR's bias can be potentially large, depending on the choice of the kernel, and choosing higher rank increases  $c_{\mathrm{PCR}}$ , thus requiring larger sample sizes. Therefore, even in well-conditioned problems (self-adjoint operator) the spurious eigenvalues may arise purely from the learning method. To facilitate detection of such occurrences, we further provide an empirical estimator of the bias of both methods and illustrate their use experimentally in Section 7.
Theorem 4. Under the assumptions of Theorems  $\boxed{2}$  and  $\boxed{3}$ , there exists a constant  $c > 0$ , depending only on  $\mathcal{H}$ , such that for large enough  $n \geq r$  and every  $i \in [r]$  with probability at least  $1 - \delta$  in the i.i.d. draw of  $\mathcal{D}_n$  from  $\rho$
$$
\left| \widehat {\lambda} _ {i} - \mu_ {j (i)} \right| \leq \left\{ \begin{array}{l l} \widehat {\eta} _ {i} \sigma_ {r + 1} \left(\widehat {C} ^ {- 1 / 2} \widehat {T}\right) + c \varepsilon_ {n} ^ {\star} \ln \delta^ {- 1}, & \widehat {G} = \widehat {G} _ {r, \gamma} ^ {\text {R R R}}, \\ \widehat {\eta} _ {i} \sqrt {\sigma_ {r + 1} (\widehat {C})} + c \varepsilon_ {n} ^ {\star} \ln \delta^ {- 1}, & \widehat {G} = \widehat {G} _ {r, \gamma} ^ {\mathrm {P C R}}. \end{array} \right. \tag {18}
$$
We remark that when  $A_{\pi}$  is of finite rank  $r$ , the bound above for the RRR estimator reduces to
$$
\left| \widehat {\lambda} _ {i} - \mu_ {i} \right| \leq c \varepsilon_ {n} ^ {\star} \ln \delta^ {- 1} \text {a n d} \| \widehat {f} _ {i} - f _ {i} \| ^ {2} \leq \frac {2 c \varepsilon_ {n} ^ {\star} \ln \delta^ {- 1}}{\left[ \operatorname {g a p} _ {i} \left(\widehat {G} _ {r , \gamma} ^ {\mathrm {R R R}}\right) - 3 c \varepsilon_ {n} ^ {\star} \ln \delta^ {- 1} \right] _ {+}},
$$
see Cor.  $\boxed{1}$  in Appendix E. Hence, in this case RRR algorithm can learn all the eigenvalues and eigenfunctions of  $A_{\pi}$  with rate  $\varepsilon_n^\star = n^{-\frac{\alpha}{2(\alpha + \beta)}}$ . On the other hand, even in this case, the bounds for the PCR estimator do not guarantee unbiased estimation of Koopman eigenvalues and eigenfunctions.
Choosing  $\gamma$  and  $r$ . The bias term  $\sigma_{r+1}(A_{\pi}S) / \sigma_r(A_{\pi}S)$  appearing in (17) represents the theoretical limit when estimating eigenvalues using RRR. It reflects the capacity of the RKHS to detect the separation of the leading  $r$  Koopman eigenvalues from the rest of its spectra. If  $A_{\pi}$  has infinite rank and slowly decaying eigenvalues, estimating the leading ones becomes challenging, since increasing  $r$  leads to smaller operator norm error, but larger bias. Luckily, in many practical problems there is a separation of time-scales in the dynamics and the above ratio can be controlled by choosing  $r$  appropriately. While we do not have access to  $A_{\pi}S$ , we can still choose  $r$  via the empirical operator  $\hat{C}^{-1/2}\hat{T}$ , see Proposition 20 of Appendix D.4. Note also that the optimal  $\gamma$  depends on  $\alpha$  which is typically unknown. In practice, one can implement a standard grid-search CV procedure for time series to tune this parameter.
Spectral Bias as a Tool for Model Selection. In equation (18), the data dependent quantities  $\widehat{s}_i(\widehat{G}_{r,\gamma}^{\mathrm{RRR}}) \coloneqq \widehat{\eta}_i\sigma_{r+1}(\widehat{C}^{-1/2}\widehat{T})$  and  $\widehat{s}_i(\widehat{G}_{r,\gamma}^{\mathrm{PCR}}) \coloneqq \widehat{\eta}_i\sigma_{r+1}(\widehat{C})$  represent the empirical spectral biases of RRR and PCR estimators of the Koopman operator, respectively. When they are small enough, the spectral estimation error is dominated by the same variance term, which decreases as the number of samples grows. Therefore, given a number of different kernels, we propose to select the best one (w.r.t. spectral estimation) by choosing the smallest spectral bias. This is illustrated in the Alenine Dipeptide example of following section.
Normal operators. Since Davis-Kahan theorem [17] also holds for normal operators, the results in this section apply whenever  $A_{\pi}A_{\pi}^{*} = A_{\pi}^{*}A_{\pi}$ . While in this case Koopman eigenfunctions remain orthogonal in  $L_{\pi}^{2}(\mathcal{X})$ , the eigenvalues are in general complex. On the other hand, extension beyond normal compact operators asks for involved spectral perturbation analysis and a new statistical learning theory.
Figure 2: Estimated eigenfunctions  $\widehat{\psi}_i$  of a Langevin dynamics vs. ground truth. The average empirical biases  $\widehat{s}_i$ ,  $i \in [4]$  are discussed at the end of Section 6. The results correspond to 50 independent estimations on 2000 training points each. PCR and RRR estimators were fitted with the same parameters: Gaussian kernel of length scale 0.175,  $\gamma = 10^{-5}$  and  $r = 4$ .
We illustrate various aspects of our theory with simple experiments. They have been implemented in Python using the library Kooplearn (available at https://github.com/CSML-IIT-UCL/kooplearn) to fit the PCR and RRR estimators. Full details are in Appendix F.
Learning the Spectrum of the Ornstein-Uhlenbeck Process. In this experiment we designed three different kernel functions (the "good", the "bad" and the "ugly") to illustrate how an unseemly kernel choice can induce catastrophic biases in the estimation of Koopman eigenvalues. We focus on the uniformly sampled Ornstein-Uhlenbeck (OU) process, discussed in Example 3, relying on the spectral decomposition of its Koopman operator  $(\mu_i, f_i)_{i \in \mathbb{N}}$  to design the three kernel functions. The good kernel is just the sum of the leading  $T = 53$  terms of the spectral decomposition of  $A_\pi$ , i.e.,  $k_{\mathrm{good}}(x, y) \coloneqq \sum_{i=1}^{T} \mu_i f_i(x) f_i(y)$ . The associated RKHS coincides with the leading eigenspace of  $A_\pi$ , and no deformation of the metric structure takes place, so that the injection map  $S \colon \mathcal{H} \hookrightarrow L_\pi^2(\mathcal{X})$  is a partial isometry. The bad kernel is defined according to the construction presented in Example 3 for  $\nu = 1/r^2$  where  $r$  is the rank of the estimator. For this kernel, the introduced bias is innocuous for RRR, but lethal for PCR. Finally, the ugly kernel corresponds to  $\nu = r^2$ , introducing large quotients  $\sigma_{r+1}(A_\pi S)/\sigma_r(A_\pi S)$  and  $\sigma_{r+1}(S)/\sigma_r(S)$ , and, hence, an irreparable bias in both estimators.
Figure 1 depicts the distribution of the eigenvalues estimated by PCR and RRR over 50 independent simulations, against the ground truth. For both algorithms each simulation is comprised of 20000 training points, the regularization is  $\gamma = 10^{-4}$  and the rank is  $r = 3$ . The three largest eigenvalues of  $A_{\pi}$  are correctly estimated by both algorithms for  $k_{\mathrm{good}}$  and by RRR for  $k_{\mathrm{bad}}$ . On the contrary, the distribution of the eigenvalues for  $k_{\mathrm{ugly}}$  (and  $k_{\mathrm{bad}}$  for PCR) does not concentrate around any true eigenvalue of  $A_{\pi}$ , signaling the presence of spurious eigenvalues in the estimation.
Figure 3: Forecasting RMSE on the Alanine Dipeptide dataset for 19 different RRR estimators, each corresponding to a different kernel, which show how the best model, according to the empirical spectral bias metric, also attains the best forecasting performances by a large margin.
A Realistic Example: Langevin Dynamics. Because of its ubiquitous use in modelling real systems, we now study a numerical implementation of the Langevin dynamics Example [1] with  $\beta = 1$  and a potential  $U(x) = 4(x^{8} + 0.8e^{-80x^{2}} + 0.2e^{-80(x - 0.5)^{2}} + 0.5e^{-40(x + 0.5)^{2}})$  that is a mixture of three Gaussians barriers at  $x\in \{-0.5,0,0.5\}$  and a smooth "bounding" term  $\propto x^8$  constraining most of the equilibrium distribution in the interval  $[-1,1]$ , see [37]. In Figure [2] for  $i\in [r]$ , we compare the  $\widehat{f}_i$  estimated by PCR and RRR against the ground truth  $f_{i}$ . The visible difficulty of PCR compared to RRR in estimating eigenfunctions is nicely explained by larger values of the empirical bias for PCR, which we report in the upper part of the figure. The reference eigenpairs of  $A_{\pi}$  have been obtained by diagonalizing a finely discretized approximation of the infinitesimal generator (see Appendix A).
Spectral Bias and Model Selection: the Case of Alanine Dipeptide. In this example we show that minimizing the first term on the r.h.s. of (18) over a validation dataset, is also a good criterion for Koopman model selection. We use a realistic simulation of the small molecule Alanine Dipeptide already discussed in [24, 44]. We trained 19 RRR estimators each corresponding to a different kernel and then we evaluated the forecasting RMSE on 2000 initial conditions drawn from a test dataset. In Figure 3 we report these errors, highlighting the model with the smallest average empirical spectral bias (18) evaluated on 5000 validation points.
We established minimax optimal rates for the operator norm error in the Koopman regression problem, which we then used to derive sharp estimation bounds for eigenvalues and eigenfunctions of the Koopman operator associated with a time-invariant Markov chain. We considered two important estimators that implement either principal component regression (PCR) or reduced rank regression (RRR) to learn a linear operator on a reproducing kernel Hilbert space. Our bounds indicate that RRR may be advantageous over PCR (also known as EDMD, the de-facto estimator in the data-driven dynamical system literature) which may exhibit a larger estimation bias. This ultimately depends on the choice of the kernel, which significantly impacts the rate. A bad choice of the kernel could also introduce spurious eigenvalues, a phenomena which has been observed in the literature and which is now explained by our theory. Finally, we proposed a method to detect spuriousness in practice, which can be used also as a kernel selection tool. A limitation of this work is that it applies to compact normal operators only. While many real dynamical systems involve such operators, in the future our analysis may be extended using more sophisticated spectral perturbation theory.
Acknowledgements. This work was supported in part from the PNRR MUR Project PE000013 CUP J53C22003010006 "Future Artificial Intelligence Research (FAIR) ", funded by the European Union - NextGenerationEU, and EU Project ELIAS under grant agreement No. 101120237.
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