  In this article we consider means of positive bounded linear operators on a
Hilbert space. We present a complete theory that provides a framework which
extends the theory of the Karcher mean, its approximating matrix power means,
and a large part of Kubo-Ando theory to arbitrary many variables, in fact, to
the case of probability measures with bounded support on the cone of positive
definite operators. This framework characterizes each operator mean
extrinsically as unique solutions of generalized Karcher equations which are
obtained by exchanging the matrix logarithm function in the Karcher equation to
arbitrary operator monotone functions over the positive real half-line. If the
underlying Hilbert space is finite dimensional, then these generalized Karcher
equations are Riemannian gradients of convex combinations of strictly
geodesically convex log-determinant divergence functions, hence these new means
are the global minimizers of them, in analogue to the case of the Karcher mean
as pointed out. Our framework is based on fundamental contraction results with
respect to the Thompson metric, which provides us nonlinear contraction
semigroups in the cone of positive definite operators that form a decreasing
net approximating these operator means in the strong topology from above.
