Pontryagin-Krein Theorem: Lomonosov's proof and related results

Abstract

We discuss Lomonosov's proof of the Pontryagin-Krein Theorem on invariant maximal nonpositive subspaces, prove the refinement of one theorem from [22] on common fixed points for a group of fractional-linear maps of operator ball and deduce its consequences. Some Burnside-type counterparts of the Pontryagin-Krein Theorem are also considered.