Analytic operator inverses are characterized near isolated singularities

What the study found: The study characterizes the inverse of an analytic Fredholm operator-valued function near an isolated singularity in a Banach space. It also describes the inverse’s Laurent series expansion in terms of the function’s Taylor coefficients around the singularity.
Why the authors say this matters: The authors conclude that these results can be applied to characterize the solution of a general autoregressive law of motion in a Banach space.
What the researchers tested: The researchers worked in a general Banach space framework and used sequential factorization of the operator function through Fredholm quotient operators. They analyzed the properties of these quotient operators near the singular point z = z0.
What worked and what didn't: The approach fully characterizes the inverse of A(z) near the isolated singularity, according to the abstract. It also provides a characterization of the Laurent series expansion of the inverse in terms of Taylor coefficients.
What to keep in mind: The abstract does not describe limitations, comparisons with other methods, or conditions beyond the isolated singularity setting in a Banach space framework.

Key points

• The paper studies an analytic Fredholm operator-valued function near an isolated singularity.
• It characterizes the inverse of the operator function in a Banach space framework.
• The inverse’s Laurent series is expressed in terms of Taylor coefficients around the singularity.
• The method uses sequential factorization through Fredholm quotient operators.
• The authors say the results can be applied to a general autoregressive law of motion in a Banach space.