Fu\v{c}\'{\i}k spectrum for discrete systems: curves and their tangent lines

TL;DR

This paper analyzes the Fu
0dedk spectrum of matrices, characterizing the existence and properties of Fu
0dedk curves, including their directions, smoothness, and tangent lines, extending previous results to non-symmetric cases.

Contribution

It generalizes the understanding of Fu
0dedk curves for matrices without symmetry assumptions, including cases with multiple eigenvalues and non-smooth curves.

Findings

Number of Fu
0dedk curves can exceed eigenvalue multiplicity.

Characterization of directions of Fu
0dedk curves emanating from eigenvalues.

Analysis of curve smoothness and tangent lines when eigenvalue multiplicity exceeds geometric multiplicity.

Abstract

In this paper, we study the Fu\v{c}\'{\i}k spectrum of a square matrix $A$ and provide necessary and sufficient conditions for the existence of Fu\v{c}\'{\i}k curves emanating from the point $(\lambda,\lambda)$ with $\lambda$ being a real eigenvalue of $A$. We extend recent results by Maroncelli (2024) and remove his assumptions on symmetry of $A$ and simplicity of $\lambda$. We show that the number of Fu\v{c}\'{\i}k curves can significantly exceed the multiplicity of $\lambda$ and determine all the possible directions they can emanate in. We also treat the situation when the algebraic multiplicity of $\lambda$ is greater than the geometric one and show that in such a case the Fu\v{c}\'{\i}k curves can loose their smoothness and provide the slopes of their "one-sided tangent lines". Finally, we offer two possible generalizations: the situation off the diagonal and Fu\v{c}\'{\i}k spectrum of a general Fredholm operator on the Hilbert space with a lattice structure.