Element of Spectrum of Self-Adjoint 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 self-adjoint densely-defined linear operator.

Let $\map \sigma T$ be the spectrum of $\struct {\map D T, T}$.

Let $\lambda \in \map \sigma T$.

Then $\lambda$ is an approximate eigenvalue of $\struct {\map D T, T}$.

Proof
Let $\lambda \in \map \sigma T$.

From Partition of Spectrum of Densely-Defined Linear Operator, $\lambda$ is contained in one of the point spectrum, continuous spectrum or residual spectrum of $T$.

From Self-Adjoint Densely-Defined Linear Operator has Empty Residual Spectrum, the residual spectrum of $T$ is empty, so $\lambda$ is either in the point spectrum or continuous spectrum.

If $\lambda$ is contained in the point spectrum, we have the result from Eigenvalue of Densely-Defined Linear Operator is Approximate Eigenvalue.

Otherwise, $\lambda$ is contained in the continuous spectrum, in which case $\paren {T - \lambda I}^{-1}$ is not bounded.

That is, there exists a sequence $\sequence {y_n}_{n \mathop \in \N}$ in $\map D {\paren {T - \lambda I}^{-1} }$ with $\norm {y_n} = 1$ and:


 * $\norm {\paren {T - \lambda I}^{-1} y_n} \to \infty$

Now set:


 * $\ds x_n = \paren {T - \lambda I}^{-1} y_n$

for each $n \in \N$ and then:


 * $\ds z_n = \frac {x_n} {\norm {x_n} }$

We then have:

Then:

so we have:


 * $\paren {T - \lambda I} z_n \to 0$

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