ELA, Volume 9, pp. 67-92, May 2002, abstract.

Shells of matrices in indefinite inner product spaces

Vladimir Bolotnikov, Chi-Kwong Li, Patrick R. Meade, 
Christian Mehl, and Leiba Rodman

The notion of the shell of a Hilbert space operator, 
which is a useful generalization (proposed by Wielandt) 
of the numerical range, is extended to operators in 
spaces with an indefinite inner product. For the most 
part, finite dimensional spaces are considered. 
Geometric properties of shells (convexity, boundedness, 
being a subset of a line, etc.) are described, as well 
as shells of operators in two dimensional indefinite 
inner product spaces. For normal operators, it is 
conjectured that the shell is convex and its closure 
is polyhedral; the conjecture is proved for indefinite 
inner product spaces of dimension at most three, and for 
finite dimensional inner product spaces with one 
positive eigenvalue.