I'm refreshing functional analysis from "Hilbert space operators in quantum physics" by Blank, Exner and Havlíček (Springer, 2008). They define essential spectrum for any closed operator on a Hilbert space as
[consisting] of all $λ ∈ \mathbb{C}$ to which there is a sequence of unit vectors $x_n ∈ D_T$ which has no convergent subsequence and satisfies $(T-λ)x_n → 0$.
I'm not sure where this definition comes from but it suits me well, other definitions I've found are valid for self-adjoint operators and for general bounded operators but either is in one way or another too restrictive. (Of course, for bounded self-adjoint operators all the definitions reduce to exactly the same thing.)
Then there is in the same source the proposition 4.7.13:
Let $A ∈ \cal{L}_{cs}(H)$. A complex number $λ$ belongs to $σ_\text{ess}(A)$ iff at least one of the following conditions is valid:
(i) $λ$ is an eigenvalue of infinite multiplicity.
(ii) The operator $(A-λ)^{-1}$ is unbounded.
On the other hand, we know that
$\sigma_\text{ap}(A) = \left\{ λ ∈ \mathbb{C} \mid (A - λ)^{-1}: Ran(A-λ) → D_A\ \text{does not exist or is not bounded} \right\}$,
which would be almost the same condition, just including also eigenvalues of finite multiplicities. This requires a Hilbert space, no mention is made of the conditions of $\cal{L}_{cs}(H)$, i.e., densely defined, closed, symmetric operators, at least not in here.
I'm happy to accept the restriction to densely defined and closed operators. All I am wondering is why the 4.7.13 is formulated only for symmetrical operators, and whether the claim holds if this condition is removed. I can't see why it should't, but I also checked the original (Czech) version of the book and there (labelled 8.4.2) they do some more magic involving the reduction of the operator $A$ to the orthogonal complement of $Ker(A-λ)$, which is defined in the English version but not really used in the proposition. The Czech version has $A_λ$ in the (ii) part, this may actually be a typo in the translation. Most importantly, if it's not the same, then also the similarity between the conditions for $σ_\text{ess}$ and $σ_\text{ap}$ breaks.
Unfortunately the book in question does not make any mention of the approximate point spectrum, so I can't compare the appropriate definitions and claims in the same language, so to say. Any clarification on the topic, or a different source where these concepts are discussed using an equivalent definition of the essential spectrum, would be most appreciated.