Commuting Unitary Hilbert Space Extensions of a Pair of Krein Space Isometries and the Krein–Langer Problem in the Band

Necessary and sufficient conditions in order that two Krein space isometries whose defect subspaces are Hilbert spaces admit commuting unitary Hilbert space extensions are discussed. Three particular cases are considered. In each of them a description of the set of commuting unitary Hilbert space extensions is presented and uniqueness conditions are given as well. The results are applied to the Krein–Langer problem in the band.