Some applications of Tanahashi's result on the best possibility of Furuta inequality

Abstract

We shall give some applications of Tanahashi’s result which states the best possibility of Furuta inequality. Firstly, we shall discuss the best possibility of a well-known characterization of chaotic order: log A log B if and only if Ar (A r 2 BpA r 2 ) r p+r holds for all p 0 and r 0 . Secondly, we shall discuss the best possibility of p-hyponormality of generalized Aluthge transformation Ts,t = |T|sU|T|t for p-hyponormal or log-hyponormal operator T whose polar decomposition is T = U|T| . Mathematics subject classification (1991): 47B20, 47A63.