Shells of matrices in indefinite inner product spaces

Abstract

Abstract. 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.

Selfadjoint operators in S-spaces

“…Spaces with an inner product given by an arbitrary bounded selfadjoint operator were studied, e.g., in [16,22]. For applications we refer to [4,6,[8][9][10][11][12]15,[17][18][19][20]26,27].…”

Normal Matrices in Degenerate Indefinite Inner Product Spaces

“…We illustrate this by help of an example. and this definition has later been taken up in [4,17]. (This way minimizes the number of times the Moore-Penrose generalized inverse of H appears in the defining equation of H-normal matrices.)…”