Primal-dual splittings as fixed point iterations in the range of linear operators

In this paper we study fixed point iterations of quasinonexpansive mappings in the range of monotone self-adjoint linear operators, which defines a real Hilbert space. This setting appears naturally in primal-dual algorithms for solving composite monotone inclusions including, as a particular instance, the Douglas--Rachford splitting. We first study conditions under which the range of a monotone self-adjoint linear operator endowed with the corresponding positive semidefinite inner product defines a Hilbert subspace, generalizing the non-standard metric case in which the linear operator is coercive and its range is the whole space. Next we study the convergence of fixed point iterations in this Hilbert subspace as a shadow of iterates in the whole Hilbert space. The result is applied to obtain the convergence of primal-dual splittings for critical values of stepsizes, generalizing results obtained in \cite{condat}. We study in detail the case of the Douglas--Rachford splitting, which is first interpreted as a primal-dual algorithm with critical stepsize values whose associated operator is firmly nonexpansive in a Hilbert subspace. A second primal-dual interpretation is provided with an alternative operator which is firmly quasinonexpansive in the whole primal-dual space with a non-standard metric. We thus obtain the weak convergence of primal-dual shadow sequences as in \cite{BausMoursi,svaiter}. We finish with some numerical experiences and applications in image processing.