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

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

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

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

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:

$\ds \paren {T - \lambda I} z_n$

$=$

$\ds \frac {\paren {T - \lambda I} x_n} {\norm {x_n} }$

$\ds $

$=$

$\ds \frac {y_n} {\norm {x_n} }$

Then:

$\ds \norm {\paren {T - \lambda I} z_n}$

$=$

$\ds \frac {\norm {y_n} } {\norm {x_n} }$

$\ds $

$=$

$\ds \frac 1 {\norm {\paren {T - \lambda I}^{-1} y_n} }$

$\ds $

$\to$

$\ds 0$

from Reciprocal of Null Sequence: Corollary, since $\norm {\paren {T - \lambda I}^{-1} y_n} \to \infty$

so we have:

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

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

$\blacksquare$

2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $25.3$: The Spectrum of Closed Unbounded Self-Adjoint Operators