Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Abstract

Consider a finite dimensional complex Hilbert space H, with dim(H) ≥ 3, define S(H):= {x ∈ H {pipe} ∥x∥ = 1}, and let νH be the unique regular Borel positive measure invariant under the action of the unitary operators in H, with νH (S(H)) = 1. We prove that if a complex frame function f: S(H) → ℂ satisfies f∈ L2(S(H), νH), then it verifies Gleason's statement: there is a unique linear operator A: H → H such that f(u) = 〈u{pipe}Au〉 for every u ∈ S(H). A is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori. © 2012 Springer Basel.