New Demiclosedness Principles for (firmly) nonexpansive operators

TL;DR

This paper extends the demiclosedness principle using orthogonal affine subspaces, providing new versions for nonexpansive operators and simplifying the proof of the Douglas-Rachford algorithm's convergence.

Contribution

It introduces generalized demiclosedness principles involving orthogonal subspaces and applies them to establish weak convergence of the Douglas-Rachford splitting method.

Findings

Extended demiclosedness principles for nonexpansive operators

New versions applicable to finitely many operators

Simplified proof of Douglas-Rachford convergence

Abstract

The demiclosedness principle is one of the key tools in nonlinear analysis and fixed point theory. In this note, this principle is extended and made more flexible by two mutually orthogonal affine subspaces. Versions for finitely many (firmly) nonexpansive operators are presented. As an application, a simple proof of the weak convergence of the Douglas-Rachford splitting algorithm is provided.