Jun Ichi Fujii

Operator Means and the Relative Operator Entropy

The notion of operator monotone functions was introduced by Lowner 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.

On the Ando–Li–Mathias mean and the Karcher mean of positive definite matrices

In this paper, from the viewpoint of the Ando–Hiai inequality, we make a comparison among three geometric means: The Ando–Li–Mathias geometric mean, the Karcher mean and the chaotic geometric mean of positive definite matrices. Among others, we show complements of the -variable Ando–Hiai inequality for the Ando–Li–Mathias geometric mean by means of the Kantorovich constant.

Continuously differentiable means

We consider continuously differentiable means, say-means. As for quasi-arithmetic means, we need an assumption that has no stationary points so that might be continuously differentiable. Introducing quasi-weights for-means would give a satisfactory explanation for the necessity of this assumption. As a typical example of a class of-means, we observe that a skew power mean is a composition of power means if is an integer.

Kolmogorov's complexity for positive definite matrices

Abstract Based on Kolmogorovs idea, complexity of positive definite matrices with respect to a unit vector is defined. We show that the range of the complexity coincides with the logarithm of its spectrum and the order induced by the complexity is equivalent to the spectral one. This order implies the reversed one induced by the operator entropy.

BOUNDS FOR AN OPERATOR CONCAVE FUNCTION

Letandbe unital positive linear maps satisfying some conditions with respect to positive scalarsand �. It is shown that if a real valued function f is operator concave on an interval J, then � (f(�(A)) �(f(A))) � f(�(A)) �(f(A)) � �(f(�(A)) �(f(A))) for every selfadjoint operator A with spectrum �(A) � J. Moreover, an external version of estimates above is presented.

Norm inequalities for matrix geometric means of positive definite matrices

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.

Operator inequalities of Malamud and Wielandt

Abstract We show that the Wielandt operator inequality and the Malamud one are equivalent and discuss some variations of them. From this point of view, we give also a proof of Malamuds multivariable inequality with its variations.

Operator means and range inclusion

We prove the range inclusions ranA12>+ranB12⊇ran(AmB)1>2>⊇ranA1>2∩ ranB1>2> for an operator mean m. An equality condition in each inclusion is given by using the representing function ƒ(x)=1 mx and F(x)=xm 1. Further we show that an operator mean m is majorized by a geometric mean if and only if ran(AmB)⊆ ranA12∩ranB12.

The unique solution of the Karcher equation and the self-adjointness of the Karcher mean

ABSTRACT Lawson and Lim showed that the Karcher equation for positive invertible operators on a Hilbert space has a unique solution using the method of the implicit function theorem of a Banach space. In this paper, in the framework of the operator inequality, we show the equivalence of the unique solution of the Karcher equation and the self-adjointness of the Karcher mean. For this, we reform the notion of the operator power means of negative order by virtue of the Tsallis relative operator entropy of negative order.

Spectrum and Entropy for Infinite Directed Graphs

From the viewpoint of operator theory, we discuss spectral properties for infinite directed graphs that have bounded valences. Graphs may have selfloops, but they are assumed not to have multiedges. Note that we use the transpose adjacency operator throughout this chapter by reason of this viewpoint. As a subsidiary effect, one may read this as a visual introduction to operator theory.