@article{Seo2023, 
author = {Won-Ki Seo},
title = {Fredholm inversion around a singularity: Application to autoregressive time series in Banach space},
year = {2023},
journal = {Electronic Research Archive},
volume = {31},
number = {8},
pages = {4925-4950},
keywords = {operator pencils, analytic Fredholm theorem, linear time-invariant dynamical systems, Granger-Johansen representation theorem},
url = {https://www.sciopen.com/article/10.3934/era.2023252},
doi = {10.3934/era.2023252},
abstract = {This paper considers inverting a holomorphic Fredholm operator pencil. Specifically, we provide necessary and sufficient conditions for the inverse of a holomorphic Fredholm operator pencil to have a simple pole and a second order pole. Based on these results, a closed-form expression of the Laurent expansion of the inverse around an isolated singularity is obtained in each case. As an application, we also obtain a suitable extension of the Granger-Johansen representation theorem for random sequences taking values in a separable Banach space. Due to our closed-form expression of the inverse, we may fully characterize solutions to a given autoregressive law of motion except a term that depends on initial values.}
}