Regular Dilations on Kreĭn Spaces

Abstract 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.