University of Split, Faculty of Electrical Engineering,
Mechanical Engineering and Naval Architecture
Abstract. The spectral condition of a matrix H
is the infimum of the condition numbers
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.