The Fučík spectrum seems to gain momentum among people working on spectral theory, with almost 300 articles published on this topic over the last 5 years, according to Google scholar. There exist several accounts and surveys about this topic, like this one by Schechter, but I am still unable to understand the deep significance of this notion. I must admit that I am a bit uneasy with the idea of introducing a completely new object whose characterization is a complete mess even in the case of small matrices.

It would be great if knowledgeable MO users could give me some hint.

- Can the Fučík spectrum of an operator or a matrix be used to say something relevant about the operator/matrix itself?
- Does the Fučík spectrum yield relevant information about the usual spectrum?
- Does the Fučík spectrum appear in relevant applications?
- [Very minimal requirement] Do there exist cases, apart from diagonal matrices, where the Fučík spectrum can be described in an easy way?