# Example of a quasinilpotent operator

Can anybody please give me an example of a quasinilpotent operator $T$, i.e. an operator such that $\sigma(T)=\{0\}$ on $l_2$ such that it has finite dimensional but non-trivial kernel and is not compact?

This is probably easy and well known but I just can't figure it out it and I am getting frustrated.

Thanks!

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@Theo Gluing = taking direct sum. Say, you have a quasinilpotent non-compact operator $T:\ell_2\to \ell_2$ with trivial kernel. Let $H=\ell_2\oplus \mathbb R$. Define $\tilde T:H\to H$ by $\tilde T(v,x)=(Tv,0)$. This operator has 1-dimensional kernel. – user31373 Aug 17 '12 at 1:59