= Factorial weighted right shift
{title2=$Te_j=e_{j+1}/j$}
On the <absolutely summable sequence space>, this weighted shift has $T^me_j=e_{j+m}/[j(j+1)\cdots(j+m-1)]$, so $\|T^m\|=1/m!$. The <spectral radius formula> gives radius zero. Its spectrum is exactly $\{0\}$ because it is not onto, while it has no eigenvalues: zero is excluded by injectivity, and any nonzero eigenvalue equation forces all coordinates to vanish starting at the first. Thus it is a nonzero <quasinilpotent element> of the operator algebra.
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