How Averaged is the Composition of Two Linear Projections?

AbstractProjection operators are fundamental algorithmic operators in Analysis and Optimization. It is well known that these operators are firmly nonexpansive; however, their composition is generally only averaged and no longer firmly nonexpansive. In this note, we introduce the modulus of averagedness and provide an exact result for the composition of two linear projection operators. As a consequence, we deduce that the Ogura–Yamada bound for the modulus of the composition is sharp.KEYWORDS: Averaged mappingFriedrichs anglemodulus of averagednessnonexpansive mappingOgura–Yamada boundprojectionMATHEMATICS SUBJECT CLASSIFICATION: Primary: 47H09Secondary: 65K0590C25 AcknowledgmentsThe authors thank the reviewers and the editors for careful reading and constructive comments. We also thank Dr. Andrzej Cegielski for making us aware of his recent work [Citation3] which contains complementary results.Notes1 Usually, one excludes the cases κ = 0 and κ = 1 in the study of averaged operators, but it is very convenient in this paper to allow for this case.2 We assume for convenience throughout the paper that the operators have full domain which is the case in all algorithmic applications we are aware of. One could obviously generalize this notion to allow for operators whose domains are proper subsets of X.Additional informationFundingThe research of the authors was partially supported by Discovery Grants of the Natural Sciences and Engineering Research Council of Canada.