From 8b9bb9346c217874670b0f1798ab6f1cb28fdb83 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Tue, 20 Feb 2024 12:01:07 +0100
Subject: Update

---
 pages/spectral-theory/test/basic.md | 55 ++++++++++++++++++-------------------
 1 file changed, 27 insertions(+), 28 deletions(-)

(limited to 'pages/spectral-theory')

diff --git a/pages/spectral-theory/test/basic.md b/pages/spectral-theory/test/basic.md
index b1015d1..9fa409b 100644
--- a/pages/spectral-theory/test/basic.md
+++ b/pages/spectral-theory/test/basic.md
@@ -7,34 +7,33 @@ nav_order: 2
 
 # {{ 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 %}
+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 $X$.
-- 
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