Kotaro Tanahashi

On log-hyponormal operators

AbstractLetT∈B(H) be a bounded linear operator on a complex Hilbert spaceH.T∈B(H) is called a log-hyponormal operator itT is invertible and log (TT*)≤log (T*T). Since log: (0, ∞)→(−∞,∞) is operator monotone, for 0<p≤1, every invertiblep-hyponormal operatorT, i.e., (TT*)p≤(T*T)p, is log-hyponormal. LetT be a log-hyponormal operator with a polar decompositionT=U|T|. In this paper, we show that the Aluthge transform

Isolated point of spectrum of p-quasihyponormal operators

\tilde T = |T|^s U|T|^t

A Characterization of Chaotic Order and a Problem

is

Quasinilpotent Part of class A or (p, k)-quasihyponormal Operators

\frac{{\min (s,t)}}{{s + t}}

The Furuta inequality in Banach *-algebras

. Moreover, ifmeas (σ(T))=0, thenT is normal. Also, we make a log-hyponormal operator which is notp-hyponormal for any 0<p.

The Furuta inequality with negative powers

Let A, B ∈ B(H) be invertible bounded linear operators on a Hilbert space H satisfying O ≤ B ≤ A , and let p, r, s, t be real numbers satisfying 1 < s, 0 < t < 1, t ≤ r, 1 ≤ p. Furuta showed that if 0 < α ≤ 1− t + r (p − t)s + r , then { A r 2 ( A− t 2 BpA− t 2 )s A r 2 }α ≤ A{(p−t)s+r}α. This inequality is called the grand Furuta inequality, which interpolates the Furuta inequality (t = 0) and the Ando-Hiai inequality ( t = 1, r = s ). In this paper, we show the grand Furuta inequality is best possible in the following sense: that is, if 1− t + r (p− t)s + r < α, then there exist invertible matrices A, B with O ≤ B ≤ A which do not satisfy { A r 2 ( A− t 2 BpA− t 2 )s A r 2 }α ≤ A{(p−t)s+r}α.