ELA, Volume1, pp. 1-17, abstract.

Numerical Ranges of an Operator on an Indefinite Inner Product Space

Chi-Kwong Li, Nam-Kiu Tsing, and Frank Uhlig

For $n \times n$ complex matrices $A$ and an $n \times n$ Hermitian
matrix $S$, we consider the {\it $S$-numerical range} of $A$ and the
{\it positive $S$-numerical range} of $A$ defined by
\[
W_S(A)=\left\{{\langle Av,v\rangle_S\over \langle
v,v\rangle_S}: v\in\IC^n, \langle v,v\rangle_S\ne 0\right\}
\]
and 
\[
W^+_S(A)=\left\{\langle Av,v\rangle_S: v\in\IC^n, \langle v,v\rangle_S
=1\right\},
\]
respectively, where $\langle u,v\rangle_S=v^*Su$. These sets generalize the
classical numerical range, and  they 
are closely related to the joint numerical range of three Hermitian
forms and the cone generated by it.
Using some theory of the joint numerical range
we can give a detailed description of $W_S(A)$
and $W_S^+(A)$ for arbitrary Hermitian matrices $S$.
In particular, it is shown that $W_S^+(A)$ is always convex
and $W_S(A)$ is always $p$-convex for all $S$.
Similar results are obtained for the sets
\[
V_S(A) =\left\{{\langle Av,v\rangle \over \langle Sv,v\rangle}:
	v\in\IC^n, \langle Sv,v\rangle \ne 0\right\}, \quad
	V_S^+(A)=\left\{\langle Av,v\rangle:\,v\in\IC^n,\langle Sv,
	v\rangle=1\right\},
\]
where $\langle u,v\rangle=v^*u$.
Furthermore, we characterize those linear operators preserving
$W_S(A)$, $W_S^+(A)$, $V_S(A)$, or $V_S^+(A)$.
Possible generalizations of
our results, including their extensions to bounded linear operators on
an infinite dimensional Hilbert or Krein space, are discussed.