The algebraic operators in the range of the Aluthge transform and the translation property
LI Ze1,2, LIU Ni1, JI Guo-xing1*
(1 College of Mathematics and Information Science, Shaanxi Normal University, Xi′an 710062, Shaanxi, China; 2 College of Science, Xi′an Polytechnic University, Xi′an 710048, Shaanxi, China)
Abstract:
The algebraic operators and idempotent operators in the range of the Aluthge transform, and the translation property of the Aluthge transform are studied. Let T be a linear bounded operator on a Hilbert space H,Δ(T) be the Aluthge transform of H. It is proved that Δ(T) is an algebraic operator if and only if T is. It is also proved that Δ(T) is an idempotent operator if and only if T3=T2. When H is a finite dimensional Hilbert space, T is a normal operator if T satisfies the translation property: Δ(T+λ)=Δ(T)+λ, ∈Ｃ.
KeyWords:
Aluthge transform; algebraic operator; idempotent operator; translation property
