---
title: Test
parent: Test
grand_parent: Spectral Theory
nav_order: 2
description: >
  The
# spellchecker:words Steinhaus preimages Baire pointwise
---

# {{ page.title }}

{: .definition-title }
> Definition (resolvent operator, regular value, resolvent set, spectrum, spectral value)
>
> 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 $T$.

{: .definition-title }
> 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.

| If $R_\lambda(T)$ exists, | is densely defined | and is bounded ... | ... then $\lambda$ belongs to the |
|:-------------------------:|:------------------:|:------------------:|-----------------------------------|
|             ✗             |          -         |          -         | point spectrum $\pspec{T}$        |
|             ✓             |          ✗         |          ?         | residual spectrum $\rspec{T}$     |
|             ✓             |          ✓         |          ✗         | continuous spectrum $\cspec{T}$   |
|             ✓             |          ✓         |          ✓         | resolvent set $\rho(T)$           |

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