# Functional Autoregressive Process - Predictive Factors

Let $$X_1,...,X_N$$ be random functions on an Hilbert space $$\mathbb{H}$$, following a Functional Autoregressive process of order 1: $$X_{t+1} = \Psi X_t + \varepsilon_t$$ Where $$\Psi$$ is a linear bounded operator on $$\mathbb{H}$$ and $$\{\varepsilon_t\} \subset \mathbb{H}$$.

In the context of prediction for the above model, we aim to estimate $$\Psi$$.
Kargin and Onatski (2008) introduce the method of Predictive Factors.

In a nutshell, they propose to to estimate a low-rank approximation of $$\Psi$$.
In order to do so, a foundamental step consists in estimating eigenvalues and eigenfunctions of another operator $$\hat{\Phi}_\alpha$$: $$\hat{\Phi}_\alpha := \hat{C}^{-1/2}_\alpha \hat{C}_1^T \hat{C}_1 \hat{C}^{-1/2}_\alpha$$

Where $$C_1$$ is the lag-1 autocovariance, $$C_\alpha = C + \alpha I$$, $$C$$ is the autocovariance of the process and $$I$$ is the identity operator.
I have derived the following form for the estimators $$\hat{C}^{-1/2}_\alpha, \hat{C}_1, \hat{C}_1^T$$:

$$\hat{C}^{-1/2}_\alpha (x)= \sum_{j=1}^{+\infty} (\hat{\lambda}_j + \alpha)^{-1/2}\langle x, \hat{\xi}_j \rangle \hat{\xi}_j \quad x \in \mathbb{H}$$ $$\hat{C}_1 (x)= \sum_{j=1}^{N-1} \langle X_j, x \rangle X_{j+1} \quad x \in \mathbb{H}$$ $$\hat{C}_1^T (x)= \sum_{j=1}^{N-1} \langle X_{j+1}, x \rangle X_{j} \quad x \in \mathbb{H}$$

Where $$\hat{\lambda}_j$$ and $$\hat{\xi}_j$$ are the estimated eigenvalues and eigenfunctions of $$C$$.

I don't know how to proceed further and how to obtain the eigenvalues and eigenfunctions of the operator $$\hat{\Phi}_\alpha$$.

It is a rather technical question, but any help is greatly appreciated! Thank you!