Let $H= L^2[0,1]$ be the space of measurable and square integrable functions from $[0,1]$ to $\mathbb{R},$ let $(\varepsilon\_k)\_{k \in \mathbb{Z}}$ denote the iid (or strict stationary) $H$-valued time series of innovations with $E(\varepsilon\_0(t)) = 0$ and $E(\varepsilon^2(t)) = 1, \forall t,$ let $\alpha\_0 = Id$ be the identity from $H$ to $H$ and let $\alpha\_1, \alpha\_2, ...$ be linear and bounded operators from $H$ to $H.$ Further there is another time series $(X\_k)\_{k \in \mathbb{Z}}$ with 


$$X\_k = \sum\_{l=0}^\infty\alpha\_l(\varepsilon\_{k-l}), \quad \forall k,\quad a.s.$$


Do you know a paper/book which deals with the estimation of the operators (with certain rates in sense of weak consistence [convergence in Probability])? 


So far I just have found a theorem dealing with the realvalued case in "Time Series & Methods" - Theorem 8.3.1 from Brockwell & Davis. 


Thank you in advance.



Estimating operators of functional linear processes
I came across the following paper from Aue and Klepsch from January:


<https://arxiv.org/pdf/1701.00770.pdf>


