TITLE: Example for a polynomially compact operator
QUESTION [1 upvotes]: I understand to say that a bounded linear operator $T$ is called "polynomially compact" if there is a nonzero polynomial $p$ such that $p(T)$ is compact. 
Can anyone help me with examples of polynomially compact operators?

REPLY [0 votes]: I think a natural example is that of nilpotent operators. Ie: operators such that $T^n = 0$, for large $n$. You can easily construct non-compact nilpotent operators. Take the $2\times2$-matrix $A = e_{2 1}$. $T = A^{\oplus \infty}$ (an infinite direct sum of $A$-blocks) is nilpotent but not compact.
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