The relative operator entropy and the Karcher mean

The Karcher mean for positive invertible operators on a Hilbert space, which is the unique solution of the Karcher operator equation, was established by Lawson-Lim based on the geometric considerations for positive invertible operators. In this note, we extend both this mean and the equation to the non-invertible case. The key concept is the relative operator entropy. For operators whose ranges are closed, we show that the Karcher mean for the non-invertible case is the unique solution of the extended Karcher equation. As a byproduct, we can show the uniqueness of the Karcher solution for the invertible case based only on that of the power means.