Noncircular elliptic discs as numerical ranges of nilpotent operators

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.

Keywords:

• numerical range
• nilpotent operator
• quasinilpotent operator
• essential numerical range

AMS Subject Classifications::

• 47A12
• 15A60

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.