= Nonzero spectra of products in opposite orders
{title2=$\sigma(ab)\setminus\{0\}=\sigma(ba)\setminus\{0\}$}
In a unital <Banach algebra>, for $\lambda\ne0$ the existence of $R=(\lambda1-ab)^{-1}$ gives $(\lambda1-ba)^{-1}=\lambda^{-1}(1+bRa)$, as verified by multiplying on either side. Interchanging $a,b$ proves the converse. Zero may differ: for the <unilateral shift operator> $S$ on $\ell^2(\mathbb N_0)$, $S^*S=1$ has spectrum $\{1\}$ and $SS^*=1-P_0$ has spectrum $\{0,1\}$.
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