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!
