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.