---
title: Test
parent: Test
grand_parent: Spectral Theory
nav_order: 2
---

# {{ page.title }}

{% definition %}
Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$.
We write

$$
T_{\lambda} = T - \lambda = T - \lambda I,
$$

where $\lambda$ is a complex number and
$I$ is the identical operator on the domain of $T$.
If the operator $T_{\lambda}$ is injective,
that is, it has an inverse $T_{\lambda}^{-1}$
(with domain $\ran{T_{\lambda}}$),
then we call

$$
R_{\lambda}(T) = T_{\lambda}^{-1} = (T - \lambda)^{-1} = (T - \lambda I)^{-1}
$$

the *resolvent operator* of $T$ for $\lambda$.
A *regular value* of $T$ is a complex number $\lambda$ for which the resolvent $R_{\lambda}(T)$ exists,
has dense domain and is bounded.
The set of all regular values of $T$ is called the *resolvent set* of $T$ and denoted $\rho(T)$.
The complement of the resolvent set in the complex plane
is called the *spectrum* of $T$ and denoted $\sigma(T)$.
The elements of the spectrum of $T$ are called the *spectral values* of $T$.
{% enddefinition %}

{% definition Point Spectrum, Residual Spectrum, Continuous Spectrum %}
Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$.
The *point spectrum* $\pspec{T}$ is the set of all $\lambda \in \CC$
for which the resolvent $R_\lambda(T)$ does not exist.
The *residual spectrum* $\rspec{T}$ is the set of all $\lambda \in \CC$
for which the resolvent $R_\lambda(T)$ exists, but is not densely defined.
The *continuous spectrum* $\cspec{T}$ is the set of all $\lambda \in \CC$
for which the resolvent $R_\lambda(T)$ exists and is densely defined, but unbounded.
{% enddefinition %}

By definition, the sets $\pspec{T}$, $\rspec{T}$, $\cspec{T}$ and $\rho(T)$
form a partition of the complex plane.