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

ON SPECTRAL CONDITION OF J-HERMITIAN OPERATORS

Krešimir Veselić and Ivan Slapničar

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 ^-1HZ is diagonal. This number controls the sensitivity of the spectrum of H under perturbations. A matrix is called J-Hermitian if H* = JHJ 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*JZ = 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.