Takeaki Yamazaki

An expression of spectral radius via Aluthge transformation

For an operator T E B(H), the Aluthge transformation of T is defined by T = |T| 2 U|T| 1/2. And also for a natural number n, the n-th Aluthge transformation of T is defined by Tn = (T n-1 ) and T 1 = T. In this paper, we shall show lim ∥T n ∥ = r(T), n→oo where r(T) is the spectral radius.

Relations between two inequalities\((B^{\tfrac{r}{2}} A^p B^{\tfrac{r}{2}} )^{\tfrac{r}{{p + r}}} \geqslant B^r andA^p \geqslant (A^{\tfrac{p}{2}} B^r A^{\tfrac{p}{2}} )^{\tfrac{p}{{p + r}}}\) and their applications

AbstractLetA andB be positive invertible operators. Then for eachp≥0 andr≥0, two inequalities

Characterizations of logA≥logB and normaloid operators via Heinz inequality

(B^{\tfrac{r}{2}} A^p B^{\tfrac{r}{2}} )^{\tfrac{r}{{p + r}}} \geqslant B^r andA^p \geqslant (A^{\tfrac{p}{2}} B^r A^{\tfrac{p}{2}} )^{\tfrac{p}{{p + r}}}

Norm inequalities for matrix geometric means of positive definite matrices

are equivalent. In this paper, we shall show relations between these inequalities in caseA andB are not invertible. And we shall show some applications of this result to operator classes.

Upper and lower bounds, and operator monotonicity of an extension of the Petz-Hasegawa function

Let T=U|T| be the polar decomposition of an operator T. Aluthge defined an operator transformation T=|T|1/2U|T|1/2 of T which is called Aluthge transformation. In this paper, we shall discuss the numerical range of T, and show the following results: 1. (i) w(T)⩾w(T). 2. (ii) If T is an n×n matrix, then W(T)⊃W(T). 3. (iii) If N(T)⊂N(T*), then W(T)⊃W(T). Moreover, we shall obtain some applications of above results.

An Operator Transform from Class A to the Class of Hyponormal Operators and its Application

AbstractIn 1951, Heinz showed the following useful norm inequality:“If A, B≥0and X∈B(H), then ‖AXB‖r‖X‖1−r≥‖ArXBr‖holds for r∈ [0, 1].” In this paper, we shall show the following two applications of this inequality:Firstly, by using Furuta inequality, we shall show an extension of Cordes inequality. And we shall show a characterization of chaotic order (i.e., logA≥logB) by a norm inequality.Secondly, we shall study the condition under which

On upper and lower bounds of the numerical radius and an equality condition

\parallel T\parallel = \parallel \tilde T\parallel