Davis–wielandt Shells of Normal Operators

For a finite-dimensional operator A with spectrum σ(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 {(λ, |λ|) : λ ∈ σ(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 an finite-dimensional operator A, and show that the Davis–Wielandt shell cannot be used to detect normality for infinite-dimensional operators. AMS Subject Classification 47A10, 47A12, 47B15