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Characterizations of connections for positive operators. (English) Zbl 1299.47076
Summary: In this paper, we investigate the relationships among algebraic, order and topological properties of a binary operation for positive operators. A connection is a binary operation satisfying monotonicity, the transformer inequality and joint-continuity from above. We show that the joint-continuity assumption can be relaxed to some conditions that are weaker than separate-continuity. Various axiomatic characterizations of connections are obtained. We show that concavity is an important property of a connection by showing that monotonicity can be replaced by concavity or 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 means.

47B65 Positive linear operators and order-bounded operators
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47A64 Operator means involving linear operators, shorted linear operators, etc.