Commutativity, comonotonicity, and Choquet integration of self-adjoint operators

Abstract

In this work, we propose a definition of comonotonicity for elements of [Formula: see text], i.e. bounded self-adjoint operators defined over a complex Hilbert space [Formula: see text]. We show that this notion of comonotonicity coincides with a form of commutativity. Intuitively, comonotonicity is to commutativity as monotonicity is to bounded variation. We also define a notion of Choquet expectation for elements of [Formula: see text] that generalizes quantum expectations. We characterize Choquet expectations as the real-valued functionals over [Formula: see text] which are comonotonic additive, [Formula: see text]-monotone, and normalized.