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

Question

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!

Answer 1

The definition of quasi-nilpotent is that the spectrum of $A$ is zero, which is the same as $\|A^n\|^{1/n} \to 0$.

See Apostol's paper On the norm-closure of nilpotents, Rev. Roumaine Math. Pures Appl. 19 (1974), 277-282 or the later paper Apostol, C.; Foiaş, C.; Pearcy, C. That quasinilpotent operators are norm-limits of nilpotent operators revisited. Proc. Amer. Math. Soc. 73 (1979), no. 1, 61–64 for info about the closure of the nilpotent operators. For recent papers giving info about hyperinvariant subspaces for quasi-nilpotent operators, see recent papers by Foias, Pearcy, et al.