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.