Roots of the identity operator and proximal mappings: (classical and phantom) cycles and gap vectors

Abstract: Recently, Simons provided a lemma for a support function of a closed convex set in a general Hilbert space and used it to prove the geometry conjecture on cycles of projections. In this paper, we extend Simons's lemma to closed convex functions, show its connections to Attouch-Thera duality, and use it to characterize (classical and phantom) cycles and gap vectors of proximal mappings.