Abstract. For a finite-dimensional operator $A$ with spectrum $\sigma(A)$, the following conditions on the Davis--Wielandt shell $DW(A)$ of $A$ are equivalent: (a) $A$ is normal. (b) $DW(A)$ is the convex hull of the set $\{(\lambda,|\lambda |^2): \lambda\in \sigma(A)\}.$ (c) $DW(A)$ is a polyhedron. These conditions are no longer equivalent for an infinite-dimensional operator $A$. In this note, a thorough analysis is given for the implication relations among these conditions. From the main result, one can deduce the equivalent conditions (a)--(c) for a finite-dimensional operator $A$, and show that the Davis--Wielandt shell cannot be used to detect normality for infinite-dimensional operators.

AMS Subject Classification (1991): 47A10, 47A12, 47B15

Keyword(s): Davis--Wielandt shell, numerical range, spectra, operator

Received January 15, 2008, and in revised form March 14, 2008. (Registered under 6075/2009.)