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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 $λ ∈ mathbbC$ 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 $A ∈ calL_cs(H)$. A complex number $λ$ belongs to $σ_textess(A)$ 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

$sigma_textap(A) = left λ ∈ mathbbC mid (A - λ)^-1: Ran(A-λ) → D_A textdoes not exist or is not bounded right$,

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 $calL_cs(H)$, 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 $σ_textess$ and $σ_textap$ 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.