Some operator equalities involving connections and means
Given a positive operator monotone function \(f\), one can define a binary operation \((A,B)\mapsto A\sigma B\) on the set of all positive bounded operators on a Hilbert space by setting \(A\sigma B=A^{1/2}f(A^{-1/2}BA^{-1/2})A^{1/2}\). These operations were called connections by A. Kubo and T. Ando. Given a connection \(\sigma\), one can construct several related connections. For instance, \(A\sigma^\perp B=(A^{-1}\sigma B^{-1})^{-1}\). Theorem 2.1. Fix connections \(\tau,\nu\). The identity \((A\tau B)(A\sigma B)^{-1}(A\nu B)=A\sigma^\perp B\) holds for all \(A,B\) and \(\sigma\) if and only if \(\nu=\tau^\perp\). Another result of this nature is proved, and consequences deduced.
scientific article; zbMATH DE number 1044235