Idempotent linear relations

TL;DR

This paper characterizes idempotent linear relations on Hilbert spaces using subspace triplets, explores sub- and super-idempotent relations, and studies their adjoints and closures.

Contribution

It provides a comprehensive characterization of idempotent linear relations and analyzes their properties, including sub- and super-idempotent cases, adjoints, and closures.

Findings

Characterization of idempotent linear relations via subspace triplets

Analysis of sub-idempotent and super-idempotent relations

Study of adjoints and closures of idempotent relations

Abstract

A linear relation $E$ acting on a Hilbert space is idempotent if $E^2=E.$ A triplet of subspaces is needed to characterize a given idempotent: $(\mathrm{ran} \, E, \mathrm{ran}(I-E), \mathrm{dom}\, E),$ or equivalently, $(\mathrm{ker}(I-E), \mathrm{ker}\, E, \mathrm{mul} \, E).$ The relations satisfying the inclusions $E^2 \subseteq E$ (sub-idempotent) or $E \subseteq E^2$ (super-idempotent) play an important role. Lastly, the adjoint and the closure of an idempotent linear relation are studied.