Inversion of an analytic operator function through Fredholm quotients and its application

TL;DR

The paper develops a unified framework to invert analytic Fredholm operator-valued functions $A(z)$ near isolated singularities in Banach spaces by using iterated Fredholm quotient factorization. It provides a closed-form Laurant expansion of $A(z)^{-1}$ in terms of the Taylor coefficients $A_{j,z_0}$ and projection data, along with a systematic method to determine the pole order. The results yield explicit expressions for the Laurent coefficients and a practical procedure to obtain the inverse, then apply the theory to autoregressive laws of motion in Banach spaces, delivering Granger-Johansen-type representations for pole orders $d=1$ and $d=2$ with exponential decay of the moving-filter coefficients. The approach extends known Hilbert-space and finite-dimensional results to a broad Banach-space setting and clarifies the role of Fredholm quotients in inverse operator analysis, with clear implications for time-series and control applications in infinite-dimensional spaces.

Abstract

We characterize the inverse of an analytic Fredholm operator-valued function A(z) near an isolated singularity within a general Banach space framework. Our approach relies on the sequential factorization of A(z) via Fredholm quotient operators. By analyzing the properties of these quotient operators near an isolated singularity, we fully characterize the Laurent series expansion of the inverse of A(z) in terms of its Taylor coefficients around the singularity. These theoretical results are subsequently applied to characterize the solution of a general autoregressive law of motion in a Banach space.

Theorems & Definitions (15)

• Proposition 2.1
• Proposition 2.2: Fredholm factorization
• Remark 2.1
• Proposition 2.3: Fredholm quotient of order $k$
• Proposition 2.4
• Proposition 3.1
• Proposition 3.2
• Proposition 4.1
• proof : Proof of Proposition \ref{['propo1']}
• proof : Proof of Proposition \ref{['propo2']}