I had asked this question on Mathematics Stack Exchange, $2$ 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 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?

I had asked this question on Mathematics Stack Exchange, $2$ 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 which defines it as $$\sigma_\text{per}(A)=\sigma(A)\cap (s(A) + i \mathbb R)$$ 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?