I had asked [this](https://math.stackexchange.com/questions/2626310/confusion-in-definition-of-peripheral-spectrum) question on [Mathematics Stack Exchange,](https://math.stackexchange.com/) $2$ days ago but it got no response so I'm asking here.

---
If $A$ is a closed operator, then the peripheral spectrum of $A$ is defined to be all those points in the spectrum with modulus equal to the spectral radius. 

However, I came across a [paper](https://arxiv.org/pdf/1511.09020.pdf) which defines it as
$$\sigma_\text{per}(A)=s(A) \cap i \mathbb R$$ where $s(A)$ denotes the spectral bound of $A,$ i.e., $s(A)=\sup\{\text{Re } \lambda:\lambda \in \sigma(A)\}.$

My question is:  Are the two definitions equivalent? If not, then what is the reason to define it in this way?