Characterization of Jordan Vectors of Operator-Valued Functions with Applications in Differential Equations
Muhamed Borogovac
TL;DR
This work extends the classical Jordan-vector characterization from matrix polynomials to holomorphic operator-valued functions $Q(z)$ by introducing root functions and generalized Jordan vectors at zeros $\alpha$ of order $l$, and by deriving necessary and sufficient conditions via the algebraic system $\sum_{p=0}^j \binom{j}{p} Q^{(p)}(\alpha)\boldsymbol{\varphi}^{(j-p)}(\alpha)=0$ for $j=0,\dots,l-1$. The main approach yields Jordan chains for operator-valued $Q$, recovering the familiar $Q(z)=A-zI$ case and extending to rational matrix-valued functions with a discussion of zeros/poles and root functions. The paper also links these spectral constructs to applications in nonlinear differential equations, illustrating how root functions can correspond to solutions of associated systems and clarifying limitations of Jordan vectors in certain non-polynomial settings. Overall, it provides a rigorous framework for spectral analysis of operator-valued functions with practical implications for differential equations and generalized Nevanlinna-type problems.
Abstract
A well-known characterization of Jordan vectors of a matrix polynomial $L(z)$ is generalized to a characterization of Jordan vectors of the operator-valued function $Q(z)$ at an eigenvalue $α\in \mathbb{C}$. The results are then applied to solve a system of nonlinear ordinary differential equations.