Invertible products in Banach algebras I found this interesting challenge: give an example of an unital Banach algebra that contains two elements $x$ and $y$ such that $xy$ is invertible but $yx$ is not invertible.
I thought it would be kind of simple to find such an example. Now, after some time, I'm not able to come up with one. Any hint?

Let $X=\ell^1$. There are bounded linear maps $S$, $T:X\to X$ such that $S(x_1,\dots)=(x_2,\dots)$ (drop the first item) and $T(x_1,\dots)=(0,x_1,\dots)$ (add a zero in the front). Then $S$, $T$ are in the Banach algebra $B(X)$ of bounded linear maps $X\to X$. 
Compute $ST$ and $TS$.
