Abstract
We show that (1) if A is a nonzero quasinilpotent operator with ran A n closed for some n ≥ 1, then its numerical range W(A) contains 0 in its interior and has a differentiable boundary, and (2) a noncircular elliptic disc can be the numerical range of a nilpotent operator with nilpotency 3 on an infinite-dimensional separable space. (1) is a generalization of the known result for nonzero nilpotent operators, and (2) is in contrast to the finite-dimensional case, where the only elliptic discs which are the numerical ranges of nilpotent finite matrices are the circular ones centred at the origin.
Acknowledgements
This research was partially supported by the National Science Council of the Republic of China under projects NSC-99-2115-M-008-008 and NSC-99-2115-M-009-002-MY2 of the respective authors. P.Y. Wu was also supported by the MOE-ATU.