Some properties of weighted operator means and characterizations of interpolational means

The theory of operator means due to Kubo-Ando is one of the most important theories in the operator theory. They have given the axiom of operator means, but have not discussed any weighted operator means. For the problem, J.I. Fujii-Kamei and Pálfia-Petz have given algorithms to make a weighted operator mean from an arbitrary (symmetric) operator mean, separately. In this paper, firstly, we shall show that the weighted operator means due to J.I. Fujii-Kamei and Pálfia-Petz are coincide. Then we shall give the dual, orthogonal and adjoint of weighted operator means. Next, we will give characterizations of interpolational means. The interpolational means were firstly considered by J.I. Fujii-Kamei. As an application of this, we will show that the characterization of operator interpolational means is only the weighted operator power means.