藤井 淳一

Based on Kolmogorov's 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. (C) 2002 Elsevier Science Inc. All rights reserved.

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 Malamud's multivariable inequality with its variations. (C) 2001 Elsevier Science Inc. All rights reserved.

We prove the range inclusions ran A1/2 + ran B1/2 superset-or-equal-to ran(Am B)1/2 superset-or-equal-to ran A1/2 and ran B1/2 for an operator mean m. An equality condition in each inclusion is given by using the representing function f(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(Am B) superset-or-equal-to ran A1/2 and ran B1/2.