Example of a Banach Algebra where $\sigma(xy) \neq \sigma(yx)$

Question

Can you help me to construct an example of a Banach Algebra where
$\sigma(xy) \neq \sigma(yx)$
NOTE: $\sigma(x)$ denotes the spectrum of $x$

Answer 1

Thinks of an example where $xy$ is invertible but $yx$ is not. In this case $0\in\sigma(yx)$ but $0\not\in\sigma(xy)$.

Specifically, consider the Banach Algebra $B(l^2(\mathbb{N}))$ of all bounded operators from $l^2(\mathbb{N})$ to $l^2(\mathbb{N})$. If we let $\{e_n\}_{n\geq 0}$ be the standard o.n. basis for $l^2(\mathbb{N})$ and define $x,y\in B(l^2(\mathbb{N}))$ as follows:

$$x(e_n) = \begin{cases} 0 & \text{if} & n = 0 \\ e_{n-1} & \text{if} & n > 0\end{cases}$$

$$y(e_n) = e_{n+1}$$

then $x$ and $y$ satisfy the condition.