Let $A$ be a Banach algebra, and suppose that $a,b\in A$ have spectra that satisfy: $\sigma(a) \subset U$, and $\sigma(b)\subset U$, where $U$ is the open right half-plane of complex numbers with positive real part.
Is is true that $\sigma(ab)$ does not contain any element of the form $-r$ for $r\geq 0$ ?
That's obviously the case when $a$ and $b$ commute, but I can't find a counterexample in the noncommutative case.
Found a counterexample (finally!). Let $A = \left(\begin{array}{rr} 1 & 0 \\ 4 & 1\end{array}\right)$ and $B = \left(\begin{array}{rr} 1 & -1 \\ 0 & 1\end{array}\right)$, then $AB = \left(\begin{array}{rr} 1 & -1 \\ 4 & -3\end{array}\right)$. Then we have that $\sigma(A) = \{1\}$, $\sigma(B) = \{1\}$ but $\sigma(AB) = \{-1\}$.
This was done by trial and error so there is not much rhyme nor reason to my choices for coefficients beyond the fact that I wanted $A$ and $B$ to be triangular (so that I could easily control their eigenvalues).