Connections and means for positive operators on a Hilbert Space

Abstract

A connection is a binary operation assigned to each pair of positive operators on a Hilbert space satisfying monotonicity, transformer inequality and continuity from above. A mean is a normalized connection. In this work, it is shown that the continuity assumption in the definition of a connection can be relaxed. We also provide various axiomatic characterizations of connections and means, involving concavity and betweenness properties. Each operator connection gives rise to a scalar connection. In fact, there is an affine order isomorphism between connections and induced scalar connections. We give an explicit description of a general connection by decomposing connections. Structures of the set of connections are also investigated. This set is isometrically order-isomorphic, as normed ordered cones, to the set of operator monotone functions on the nonnegative reals. It is isometrically isomorphic, as normed cones, to the set of finite Borel measures on the extended half-line. Moreover, we establish some properties of connections related to operator inequalities.