Added basics of Banach algebras.

src/op/banach/invertible.tex

@@ -4,15 +4,17 @@
 
 \begin{definition}[Invertible]
 \label{definition:banach-algebra-invertible}
-    Let $A$ be a unital Banach algebra and $x \in A$, then $x$ is \textbf{invertible} if there exists $x^{-1} \in A$ such that $xx^{-1} = x^{-1}x = 1$. The set $G(A)$ denotes the collection of all invertible elements in $A$. 
+    Let $A$ be a unital Banach algebra and $x \in A$, then $x$ is \textbf{invertible} if there exists $x^{-1} \in A$ such that $xx^{-1} = x^{-1}x = 1$. The set $G(A)$ denotes the group of all invertible elements in $A$. 
 \end{definition}
 
-\begin{lemma}
+\begin{lemma}[Neumann Series]
 \label{lemma:neumann-series}
     Let $A$ be a unital banach algebra and $x \in B_A(1, 1)$, then $x \in G(A)$ with
     \[
     x^{-1} = \sum_{n = 0}^\infty (1 - x)^n
     \]
+
+    and $\normn{x^{-1}}_A \le (1 - \norm{x})_A^{-1}$. 
 \end{lemma}
 \begin{proof}
     Since $\norm{1 - x}_A < 1$, the series converges absolutely. Let $y = \sum_{n = 0}^\infty (1 - x)^n$, then
@@ -50,3 +52,27 @@
     (3): Since the inversion map is locally a power series, it is $C^\infty$ by \autoref{theorem:termwise-differentiation}. 
 \end{proof}
 
+\begin{proposition}
+\label{proposition:swap-invertible}
+    Let $A$ be a unital Banach algebra and $x, y \in A$, then $1 - xy \in G(A)$ if and only if $1 - yx \in G(A)$. 
+\end{proposition}
+\begin{proof}[Proof, {{\cite[Proposition I.3.4]{Zhu}}}. ]
+    If $1 - xy \in G(A)$, then
+    \begin{align*}
+        (1 - yx)[y(1 - xy)^{-1}x + 1] &= y(1 - xy)^{-1}x + 1 \\
+        &-yxy(1 - xy)^{-1}x - yx \\
+        &= y[(1 - xy)^{-1} - xy(1 - xy)^{-1}]x + 1 - yx \\
+        &= yx + 1 - yx = 1
+    \end{align*}
+    
+    Similarly, 
+    \begin{align*}
+        [y(1 - xy)^{-1}x + 1](1 - yx) &=  y(1 - xy)^{-1}x + 1 \\
+        &-y(1 - xy)^{-1}xyx - yx \\
+        &= y[(1 - xy)^{-1} - (1 - xy)^{-1}xy]y + 1 - yx \\
+        &= yx + 1 - yx = 1
+    \end{align*}
+    
+\end{proof}
+
+