Oh! Maybe $\sigma_{e5}$ is the right definition to make sure $\sigma(A)\setminus\sigma_{\text{ess}}(A)\subset\sigma_{\text{d}}(A)$, am I right?

@DCM Thank you every much for your comment! This book helps a lot! It lists all kinds of definitions of essential spectra, wow! Recently I'm reading a paper which uses the fact that $\sigma(A)\setminus\sigma_{\text{ess}}(A)\subset\sigma_{\text{d}}(A)$, which is well known for self-adjoint operators in Hilbert spaces . The paper says that it takes the definition as $\sigma_{e3}$ in section 4 of chapter 1 of the book you mentioned. I'm looking for a proof now, which is likely not contained in this book, right? Do you know any reference for it?