Numerical Range of the Aluthge Transform
Xiumei Liu · Acta Analysis Functionalis Applicata · 2005
For any bounded linear operator A on a Hilbert space H, if A=V｜A｜ is a polar decomposition of A, the Aluthge transform and the ()~*-Aluthge transform ~((*)) of A are defined by =|A|~1[]2 V｜A｜~(1[]2) and ~((*))=｜A~*~(12) V｜A~*~(1[]2), respectively. Denote by W() and W(~((*))) the numerical ranges of and ~((*)), respectively. In this note, we prove that W()=W(~((*))). This confirms a conjecture of Wu.