Definition:Approximate Eigenvalue of Densely-Defined Linear Operator

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.

Let $\lambda \in \C$.

We say that $\lambda$ is an approximate eigenvalue if and only if:

there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $\map D T$ such that:

$\paren {T - \lambda I} x_n \to 0$

Also see

• Eigenvalue of Densely-Defined Linear Operator is Approximate Eigenvalue
• Element of Spectrum of Self-Adjoint Densely-Defined Linear Operator is Approximate Eigenvalue

• Results about approximate eigenvalues 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