Operator Means and the Relative Operator Entropy

The notion of operator monotone functions was introduced by Löwner and that of operator concave functions by Kraus who is his student. Operator means were introduced by Ando and the general theory of them was established by Kubo and Ando himself. By their theory, a nonnegative operator monotone function is now considered as a variation of an operator mean. However this theory does not include the logarithm and the entropy function which are operator monotone and often used in information theory. These functions are operator concave and satisfy Jensen’s inequality. So, considering operator means from the historical viewpoint, we shall introduce the relative operator entropy by generalizing the Kubo-Ando theory. Though its definition is derived from the Kubo-Ando theory of operator means, it can be constructed also in some ways. The relative operator entropy has of course some entropy-like properties.