Optimal normal projections in Krein spaces

JUAN IGNACIO GIRIBET; ALEJANDRA MAESTRIPIERI; FRANCISCO MARTÍNEZ PERÍA

Opatija

Conferencia; 4th Najman Conference on Spectral Problems for Operators and Matrices; 2015

Department of Mathematics, University of Zagreb

In the previous Najman Conference we introduced the $J$-normal projections acting on a Krein space $\KK$, i.e. those (bounded) projections $Q$ in $\KK$ that commutes with its $J$-adjoint $Q^\#$. A closed subspace $\St$ of $\KK$ is the range of a $J$-normal projection if and only if $\St$ is pseudo-regular. Moreover, if the isotropic part $\St^\circ$ is non-trivial, there exist infinitely many $J$-normal projections onto $\St$ which can be parametrized according to a suitable decomposition of $\KK$.Along this talk, for a fixed pseudo-regular subspace $\St$ of $\KK$, a $J$-normal projection $Q_0$ onto $\St$ is distinguished. Its operator norm can be calculated in terms of the Friedrichs angle between $\St$ and $\St^{\ort}$ (in fact, $\|Q_0\|$ is the cosecant of this angle), and it is minimal among the norms of the $J$-normal projections onto $\St$.