Definition:Point Spectrum of Densely-Defined Linear Operator/Eigenvalue

Definition
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.

Let $\struct {\map D T, T}$ be a densely-defined linear operator.

We say that $\lambda \in \C$ is an eigenvalue of $T$ there exists $x \in \map D T \setminus \set 0$ such that:


 * $T x = \lambda x$

Also see

 * Point Spectrum of Densely-Defined Linear Operator consists of its Eigenvalues, where it is shown that these are precisely the elements of the point spectrum of $T$.