Definition:Point Spectrum of Densely-Defined Linear Operator/Eigenvalue
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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 $T$ if and only if 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 $T$.
Sources
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $25.3$: The Spectrum of Closed Unbounded Self-Adjoint Operators