Naslov
On spectral condition of J-Hermitian operators

Autori
Veselić, Krešimir ; Slapničar, Ivan

Izvornik
Glasnik matematički Memorial Issue in Honor of Branko Najman 35(55) (2000), 1; 3-24

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Hermitian operator; spectral condition number; Hilbert space;

Sažetak
The spectral condition of a matrix $H$ is the infimum of the condition numbers $\kappa(Z) = \|Z\|\|Z^{-1}\|$, taken over all $Z$ such that $Z^{-1}HZ$ 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 matrices $Z$, 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 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 (with the notion of ``diagonalization'' appropriately generalized) and are applicable even to unbounded operators. We apply our theory to the Klein--Gordon operator thus improving a previously known bound.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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