Characterizations of Connections for Positive Operators

An axiomatic theory of operator connections and operator means was investigated by Kubo and Ando in 1980. A connection is a binary operation for positive operators satisfying the monotonicity, the transformer inequality and the joint-continuity from above. In this paper, we show that the joint-continuity assumption can be relaxed to some conditions which are weaker than the separate-continuity. This provides an easier way for checking whether a given binary opertion is a connection. Various axiomatic characterizations of connections are obtained. We show that the concavity is an important property of a connection by showing that the monotonicity can be replaced by the concavity or the midpoint concavity. Each operator connection induces a unique scalar connection. Moreover, there is an affine order isomorphism between connections and induced connections. This gives a natural viewpoint to define any named means.