Exponential of a rank-one operator
= Exponential of a rank-one operator
{title2=$e^{tK}=I+\frac{e^{t\varphi(a)}-1}{\varphi(a)}K$}
For the bounded <rank-one operator> $Kx=a\varphi(x)$, insert the <powers of a rank-one operator> into the convergent <operator exponential> <power series>. Writing $c=\varphi(a)$ gives the displayed formula for $c\ne0$. At $c=0$ the quotient has <limit> $t$, so $e^{tK}=I+tK$. This includes the zero operator and the nonzero <nilpotent operator> cases without dividing by zero.