Singular unitary dilations

Abstract

We study the problem of determining which bounded linear operator on a Hilbert space can be dilated to a singular unitary operator. Same of the partial results we obtained are (1) every strict contraction has a diagonal unitary dilation, (2) every C-0 contraction has a singular unitary dilation, and (3) a contraction with one of its defect indices finite has a singular unitary dilation if and only if it is the direct sum of a singular unitary operator and a C-0(N) contraction. Such results display a scenario which is in marked contrast to that of the classical case where we have the absolute continuity of the minimal unitary power dilation of any completely nonunitary contraction.