An analytic family of operators whose Jordan chains stabilize at length k can be locally diagonalized to a diagonal operator polynomial of degree k.
Beyn, An integral method for solving nonlinear eigenvalue problems, Linear Algebra and Its Applications 436, 3839–3863, 2012
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Diagonalization of Operator functions by algebraic methods
An analytic family of operators whose Jordan chains stabilize at length k can be locally diagonalized to a diagonal operator polynomial of degree k.