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I have a question regarding operator theory and would be glad if someone could help. I have a linear operator $A$ that is non-self-adjoint, unbounded and is densely defined in a Hilbert space $H$. I know how the spectrum of this operator looks like. However, not all points of the spectrum are eigenvalues. Given a point in the spectrum $\lambda_j$ (in my case $\lambda_j \in \mathbb{R}$), is there a way to check, if this point is an eigenvalue in the mathematical meaning of this word, i.e., that there exist a function $\varphi_j \in H$ such that $A\varphi_j = \lambda_j\varphi_j$?

Of course, the operator $A$ is given explicitly, i.e., I know how the corresponding linear prescription looks like.

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If the only data you have available is the spectrum, then the answer is no. You can have two operators with the same spectrum, where a certain point is an eigenvalue for one operator and not an eigenvalue for the other.

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