Eigenvalue of Densely-Defined Linear Operator is Approximate Eigenvalue

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 $\lambda$ be an eigenvalue of $T$.

Then $\lambda$ is an approximate eigenvalue of $T$.

Proof

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

$\paren {T - \lambda I} x = 0$

Then setting:

$\ds x_n = \frac x {\norm x}$

we have:

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

for each $n \in \N$, while $\norm {x_n} = 1$.

Then, we have:

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

So $\lambda$ is an approximate eigenvalue of $T$.

$\blacksquare$