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In this article we consider means of positive operators on a Hilbert space.We extend the theory of matrix power means to arbitrary operator means in thesense of Kubo-Ando. The basis of the extension is relying on ideas coming fromdifferential geometry. We consider generalized Karcher equations for positiveoperators and show that such equations admit unique positive solutions that canbe obtained as a limit of one-parameter families of operator means calledinduced operator means. These means are themselves unique fixed points of oneparameter families of strict contractions induced, through Kubo-Ando theory ofoperator means, by semigroups of holomorphic functions mapping the upperhalf-plane into itself. These semigroups of holomorphic functions areconsidered with Koenigs function corresponding to Schroeder's functionalequation. Koenigs function in this setting provides us with a logarithm mapcorresponding to every 2-variable operator mean. The semigroups of 2-variablemeans behave as geodesics and we exactly classify the cases when they areindeed geodesics of affinely connected manifolds, thereby providing the caseswhen these generalized Karcher equations are exactly Karcher equations in thegeometric sense. This is achieved by studying the arising holonomy groups. Theunique solutions of these generalized Karcher equations are called lambdaextensions and have numerous desirable properties which are inherited from theinduced operator means themselves.

Semigroups of operator means and generalized Karcher equations