Idempotent linear 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, $(\ker (I-E),\ker E,\mathrm{mul}E).$ 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.