Characterization of Jordan Vectors of Operator-Valued Functions with Applications in Differential Equations

TL;DR

This paper generalizes the concept of Jordan vectors from matrices to operator-valued functions at eigenvalues and applies these results to solve nonlinear differential equations.

Contribution

It introduces a new characterization of Jordan vectors for operator-valued functions, extending classical matrix results to a broader functional context.

Findings

Generalized Jordan vector characterization for operator-valued functions.

Applied the theoretical results to solve nonlinear ordinary differential equations.

Provided a framework connecting operator theory with differential equations.

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 $\alpha \in \mathbb{C}$. The results are then applied to solve a system of nonlinear ordinary differential equations.