2013/04/09 by Pattrawut Chansangiam, Chansangiam, Pattrawut, Wicharn Lewkeeratiyutkul +1
Mathematics · #47A63 #47A64 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Inequalities and Applications #math.FA #msc:47A63 #msc:47A64
paper · pdf · doi:10.48550/arxiv.1304.2452
9 pages
arxiv created 2013/04/09 · openalex publication_date 2013/04/09 · arxiv updated 2013/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A connection in Kubo-Ando sense is a binary operation for positive operators on a Hilbert space satisfying the monotonicity, the transformer inequality and the continuity from above. A mean is a connection σ such that A σA =A for all positive operators A. In this paper, we consider the interplay between the cone of connections, the cone of operator monotone functions on \R+ and the cone of finite Borel measures on [0,∞]. %We define a norm for a connection in such a way that the set of operator connections becomes %a normed ordered cone. %On the other hand, the cone of operator monotone functions on \R+ %and the cone of finite Borel measures on [0,∞] are equipped with suitable norms. The set of operator connections is shown to be isometrically order-isomorphic, as normed ordered cones, to the set of operator monotone functions on \R+. This set is isometrically isomorphic, as normed cones, to the set of finite Borel measures on [0,∞]. It follows that the convergences of the sequence of connections, the sequence of their representing functions and the sequence of their representing measures are equivalent. In addition, we obtain characterizations for a connection to be a mean. In fact, a connection is a mean if and only if it has norm 1.