ELA, Volume 3, pp. 31-47, April 1998, abstract.

Shapes and Computer Generation of Numerical Ranges of Krein Space Operators

Chi-Kwong Li and Leiba Rodman


Let (K,<.,.>) be a Hilbert space equipped with an indefinite inner product 
[.,.].  Then (K,[.,.]) is a (complex) Krein space.  One can define the 
Krein space numerical range of an operator A acting on K as the collection 
of complex numbers of the form [Av,v] with v in K satisfying [v,v] = 1.
In this paper, the shapes of Krein space numerical ranges of operators in 
the complex plane using the joint numerical range of self-adjoint operators 
on (K,<.,.>) are studied.  Krein space numerical ranges of operators acting 
on a two-dimensional space are fully described. A Matlab program is 
developed to generate the sets in the finite dimensional case.