Original scientific paper

On spectral condition of J-Herminian operators

Krešimir Veselić
Ivan Slapničar

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.

Keywords

Hrčak ID:

4859

URI

https://hrcak.srce.hr/4859

Publication date:

1.6.2000.

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