# Limits of Nilpotent and Quasi-nilpotent Operators in a $\mathrm{II}_1$-factor

A bounded operator $A$ in a Hilbert space is called nilpotent if there exists $n$ such that $A^{n}=0$. An operator is called quasi-nilpotent iff $$\limsup_{n\to\infty}{ \|A^{n}\|^{1/n}}=0.$$

Every nilpotent operator is clearly quasi-nilpotent. The family of quasi-nilpotent operators is very important for the hyperinvariant subspace problem. For instance, it was proved by Haagerup and Schultz, that if the Brown measure of an operator in a $\mathrm{II}_1$-factor is concentrated in more than one point then it has a non-trivial hyperinvariant subspace. A subspace is called $A$-hyperinvariant if it is invariant for all the operators that commute with $A$.

I seem to recall from Herrero's book that the norm closure of the nilpotent and quasi-nilpotent operators in $B(H)$ is pretty well understood. However, I don't have a copy with me at the moment. My question is: Is it known what is the norm closure of the nilpotent operators or/and quasi-nilpotent operators in the hyperfinite $\mathrm{II}_1$-factor? In any other $\mathrm{II}_1$-factor?

Thanks!

-

The definition of quasi-nilpotent is that the spectrum of $A$ is zero, which is the same as $\|A^n\|^{1/n} \to 0$.
Thanks Bill! I corrected. These papers focus on the $B(H)$ case right? I'm more interested in the $\mathrm{II}_{1}$ factor situation, do you know any references there? – ght Apr 13 '11 at 21:11
Certainly their focus was on $B(H)$, Gabriel. I did not study the papers to see what all they did. Have you tried to check on math sci-net? – Bill Johnson Apr 13 '11 at 22:18