The range of the Douglas–Rachford operator in infinite-dimensional Hilbert spaces

Abstract The Douglas–Rachford algorithm is one of the most prominent splitting algorithms for solving convex optimization problems. Recently, the method has been successful in finding a generalized solution (provided that one exists) for optimization problems in the inconsistent case (i.e., when a solution does not exist). The convergence analysis of the inconsistent case hinges on the study of the range of the displacement operator associated with the Douglas–Rachford splitting operator and the corresponding minimal displacement vector. A comprehensive study of this range has been developed in finite-dimensional Hilbert spaces. In this paper, we provide a formula for the range of the Douglas–Rachford splitting operator in (possibly) infinite-dimensional Hilbert spaces under mild assumptions on the underlying operators. Our new results complement known results in finite-dimensional Hilbert spaces. Several examples illustrate and tighten our conclusions.