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Suppose $T$ is a bounded operator on $H:=\mathcal{l}_2$ which is quasinilpotent and has the property that both $T$ and $T^{*}$ are not injective and have finite dimensional kernels.

Is it possible that $T$ restricted to any infinite dimensional invariant subspace has the same property? It would be natural to look at the decreasing chain of invariant subspaces $\overline{T^{n}(H)}$ and try to prove that the restriction of $T$ to one of those fails this property (so either $T$ or $T^{*}$ will be injective). I wasn't able to show this and I suspect such operators exists. If you can give me such a concrete example, or perhaps show that it is not possible, I would appreciate it. It is probably very hard to come up with an example such that the restriction to any infinite dimensional invariant subspace has the property, so I will settle with an example when you restrcit $T$ to the "obvious" invariant subspaces above.


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