Definition:Point Spectrum of Densely-Defined Linear Operator
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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 define the point spectrum $\map {\sigma_p} T$ as the set of $\lambda \in \C$ such that:
- $T - \lambda I$ is not injective.
Eigenvalue
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
- Results about point spectrums of densely-defined linear operators can be found here.
Sources
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $25.3$: The Spectrum of Closed Unbounded Self-Adjoint Operators