Quasi-Invertibility of x When r(x) < 1 in a Banach Algebra
Quasi-Invertibility of a When r(x) < 1 in a Banach Algebra
| Theorem 1: Let $\mathfrak{A}$ be a Banach algebra and let $x \in \mathfrak{A}$. If $r(x) < 1$ then $x$ is quasi-invertible and the quasi-inverse of $x$ is $\displaystyle{x^0 = - \sum_{n=1}^{\infty} x^n}$. |
Note that we do not require $\mathfrak{A}$ to have a unit.
- Proof: Since $\mathfrak{A}$ is a Banach algebra, its unitization $\mathfrak{A} + \mathbf{F}$ is a Banach algebra with unit. Consider the point $(x, 0) \in \mathfrak{A} + \mathbf{F}$. We have that:
\begin{align} \quad r((x, 0)) &= \inf \left \{ \| (x, 0)^n \|^{1/n} : n \in \mathbb{N} \right \} \\ &= \inf \left \{ \| (x^n, 0) \|^{1/n} : n \in \mathbb{N} \right \} \\ &= \inf \left \{ (\| x^n \| + |0|)^{1/n} : n \in \mathbb{N} \right \} \\ &= \inf \left \{ \| x^n \|^{1/n} : n \in \mathbb{N} \right \} \\ &= r(x) \\ & < 1 \end{align}
- So by the theorem on the Invertibility of 1 - a When r(a) < 1 in a Banach Algebra with Unit page, since $\mathfrak{A} + \mathbf{F}$ is a Banach algebra with unit $(0, 1)$, and $r((x, 0)) < 1$ we have that since $(0, 1) - (x, 0)$ is invertible and the inverse is given by:
\begin{align} \quad [(0, 1) - (x, 0)]^{-1} &= (0, 1) + \sum_{n=1}^{\infty} (x, 0)^n \\ &= (0, 1) - \left ( - \sum_{n=1}^{\infty} (x, 0)^n \right ) \\ &= (0, 1) - \left ( - \sum_{n=1}^{\infty} (x^n, 0) \right ) \\ &= (0, 1) - \left (- \sum_{n=1}^{\infty} x^n, 0 \right ) \end{align}
- But from the proposition on the Algebras - x Has Quasi-Inverse y IFF (0, 1) - (x, 0) Has Inverse (0, 1) - (y, 0) page we have that $x$ is quasi-invertible with quasi-inverse:
\begin{align} \quad x^0 &= - \sum_{n=1}^{\infty} x^n \quad \blacksquare \end{align}