ELA, Volume 11, pp. 192-204, September 2004, abstract.

Hyponormal Matrices and Semidefinite Invariant 
Subspaces in Indefinite Inner Products

Christian Mehl, Andre C.M. Ran, and Leiba Rodman

It is shown that, for any given polynomially normal 
matrix with respect to an indefinite inner product, 
a nonnegative (with respect to the indefinite inner 
product) invariant subspace always admits an extension 
to an invariant maximal nonnegative subspace. Such an 
extension property is known to hold true for general 
normal matrices if the nonnegative invariant subspace 
is actually neutral. An example is constructed showing 
that the extension property does not generally hold 
true for normal matrices, even when the nonnegative 
invariant subspace is assumed to be positive. On the 
other hand, it is proved that the extension property 
holds true for hyponormal (with respect to the 
indefinite inner product) matrices under certain 
additional hypotheses.