For bounded operators on Kreĭn spaces, isometric or unitary dilations always exist. We prove that any minimal isometric or unitary dilation has a precise geometrical structure. Moreover, a bounded operator T has a unique minimal unitary dilation if and only if T and $T^*$ have unique minimal isometric dilation if and only if T is either contractive or expansive and $T^*$ is either contractive or expansive. Passing to the bi-dimensional case, a minimal unitary extension (in short, m.u.e.) $U=(U_1, U_2)$ is obtained for a pair $V=(V_1, V_2)$ of commuting bounded isometries on a Kreĭn space. There is a link with the one-dimensional case: if U is an m.u.e. for $V,$ then $U_1U_2$ is an m.u.e. for $V_1V_2$. Also, if $(V_1V_2)^*$ is either contractive or expansive, then V has a unique minimal unitary extension. A minimal regular isometric dilation is obtained for a commuting pair $T=(T_1, T_2)$ of bounded operators on a Kreĭn space such that $T_1,\ T_2$ are contractions and T is a bidisc contraction or $T_1,\ T_2$ are expansions and T is a bidisc expansion. The existence of a minimal unitary extension is used to provide a minimal regular unitary dilation for T. Discussions about uniqueness and geometric structure conclude the article.

We investigate the influence of interface conditions at a singularity of an indefinite canonical system on its Weyl coefficient. An explicit formula which parametrizes all possible Weyl coefficients of indefinite canonical systems with fixed Hamiltonian function is derived. This result is illustrated with two examples: the Bessel equation, which has a singular end point, and a Sturm–Liouville equation whose potential has an inner singularity, which arises from a continuation problem for a positive definite function.

Decompositions of spectral type are obtained for closed Hilbert space operators with empty resolvent set, but whose square has closure which is spectral. Krein space situations are also discussed.

A partition of the class of all Hermite–Biehler entire functions of finite order into subclasses ${\cal P}_\kappa$ is introduced. A given function $E(z)$ belongs to ${\cal P}_\kappa$ if and only if $-z^{-1}\log E(z)\in{\mc N}_\kappa$, where the class ${\mc N}_\kappa$ is a well studied family of meromorphic functions on the upper half plane, which originates from operator theoretic problems. It is also proved that the subclasses ${\cal P}_\kappa$ are stable under bounded type perturbation.