Definition:Approximate Eigenvalue of Densely-Defined Linear Operator

Theorem
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 :


 * 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
 * Self-Adjoint Densely-Defined Linear Operator has Approximate Eigenvalue