Let $\mathcal{H}$ and $\mathcal{L}({\mathcal{H}})$ denote a separable Hilbert space and the set of bounded linear operators on it respectively.
As a graduate student entering the field of non-selfadjoint operators I am asking about the distinction between normal points (a concept seemingly appearing only in older literature, see below) and discrete eigenvalues (in the usual sense: the subset of the point spectra which are isolated eigenvalues of finite algebraic multiplicity).
From the book by Gohberg and Krein [1]: $\lambda_0 \in \mathbb{C}$ is a normal point of $A \in \mathcal{L}({\mathcal{H}})$ if and only if:
- $\lambda_0$ is an isolated point of the spectrum of $A$;
- the algebraic multiplicity of $\lambda_0$ is finite;
- the set $(A-\lambda_0 I)\mathcal{H}$ of values of the operator $A-\lambda_0 I$ is closed.
Now, it is clear that $\sigma_{normal}(A) \subseteq \sigma_{discrete}(A)$.
How ''strong'' is condition 3? I mean, what easy observations you extract from condition 3, so if it is typically the case that results obtained for normal points are not applicable to discrete eigenvalues, i.e. if they are most likely to lie in the essential spectrum or not.
This question interests me because of the following spectral stability theorem which appears in the same book (at the end of Chapter 1), implying the corollary I write below:
Theorem: Let $A = U + K$ where $U$ is unitary and $K$ is compact. If the resolvent of $A$ contains at least one point in the open unit disk, then $\sigma_{normal}(A)=\sigma_{normal}(U)$.
Corollary: Let $U_1 = U_2 K$ where $U_1$ and $U_2$ are unitary operators and $K-I$ is compact. Then $\sigma_{normal}(U_1)=\sigma_{normal}(U_2)$.
[1] Introduction to linear non-selfadjoint operators, Gohberg, Krein (1969).
