Structure of the operators that commute twice with operators of class K(H) (Q1091585)

According to a theorem of J. von Neumann the double commutant of a selfadjoint operator coincides with the weakly closed algebra generated by it. The corresponding assertion, however, is not true for a J- selfadjoint operator in a Krein space with canonical involution J. Among others, the author mentions, without proof, that if A is a J-selfadjoint operator, having a maximal J-non-negative invariant subspace which is a direct sum of a one-dimensional J-neutral subspace and a uniformly J- positive subspace then the double commutant of A coincides with the weakly closed subalgebra generated by it.