= Norm stability of noninvertibility for the unilateral right shift
{title2=$\|A\|<1\Longrightarrow R+A\text{ noninvertible}$}
On $\ell_1$, the <unilateral shift operator> $R$ and <left shift operator> $L$ satisfy $LR=I$ and $\|L\|=1$, while $L$ has nontrivial kernel. If $\|A\|<1$, the <Neumann series> makes $L(R+A)=I+LA$ invertible. Were $R+A$ invertible, this identity would express $L$ as a product of invertible operators, a contradiction. Thus the open unit ball about $R$ contains no invertible operator.
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