Example for a polynomially compact operator

Question

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?

Answer 1

Every compact operator is polynomially compact.

Edit. If you want to have a non-trivial example take any bounded non-compact operator $T$ such that $T^2$ is compact (see e.g. this answer). Then one can take $p(x) = x^2$.

Answer 2

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.