Let A be a quasinilpotent operator on a Hilbert space and let every operator of the algebra generated by $A$ and $A^{*}$ have finite spectrum. Does then follow, that A is nilpotent ?
See also http://mathoverflow.net/questions/77177/quasinilpotence-and-finite-spectrum and http://mathoverflow.net/questions/35207/finite-dimensional-subalgebras-of-c-star-algebras
