#### Glasnik Matematicki, Vol. 35, No.1 (2000), 3-23.

### ON SPECTRAL CONDITION OF *J*-HERMITIAN
OPERATORS

### Krešimir Veselić and Ivan Slapničar

Fernuniversitat Hagen, Lehrgebiet Mathematische Physik,
Postfach 940, D-58084 Hagen, Germany

*e-mail:* `Kresimir.Veselic@FernUni-Hagen.de`
University of Split, Faculty of Electrical Engineering,
Mechanical Engineering and Naval Architecture

*e-mail:* `Ivan.Slapnicar@fesb.hr`

**Abstract.** The spectral condition of a matrix *H*
is the infimum of the condition numbers
κ(*Z*) =
||*Z*|| ||*Z*^{ -1}||, taken over all *Z*
such that *Z*^{ -1}*H**Z* is diagonal.
This number controls the sensitivity of the spectrum of *H*
under perturbations. A matrix is called *J*-Hermitian if
*H** = *J**H**J* for some
*J* = *J** = *J*^{ -1}.
When diagonalizing *J*-Hermitian matrices it is natural to look
at *J*-unitary *Z*, that is, those that satisfy
*Z***J**Z* = *J*.

Our first result is: if there is such *J*-unitary *Z*,
then the infimum above is taken on *J*-unitary *Z*,
that is, the *J* unitary diagonalization is the most stable of
all. For the special case when *J*-Hermitian matrix has
definite spectrum, we give various upper bounds for the spectral
condition, and show that all *J*-unitaries *Z* which
diagonalize such a matrix have the same condition number.
Our estimates are given in the spectral norm and the
Hilbert-Schmidt norm. Our results are, in fact, formulated and proved
in a general Hilbert space (under an appropriate generalization of the
notion of 'diagonalising') and they are applicable even to unbounded
operators. We apply our theory to the Klein-Gordon operator thus
improving a previously known bound.

**1991 Mathematics Subject Classification.**
15A12, 65F35, 15A57, 15A60.

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