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arXiv · 1208.4912

Characterizations of Connections for Positive Operators

Pattrawut Chansangiam·Wicharn Lewkeeratiyutkul
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Abstract

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.

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Pattrawut Chansangiam, Wicharn Lewkeeratiyutkul. 2012-08-24. Characterizations of Connections for Positive Operators. https://arxiv.org/abs/1208.4912

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