Numerical range of Aluthge transform of operator
A bounded linear operator \(A\) on a complex Hilbert space \(H\) has a unique polar decomposition \(A=V|A|\), where \(|A|= (A^*A)^{1 \over 2}\) and \(V\) is a partial isometry. Associated with \(A\), there is a very useful related operator \(\widetilde A=|A|^{1\over 2}V|A|^{1 \over 2}\), called the Aluthge transform of \(A\) [see \textit{A. Aluthge}, Integral Equ. Oper. Theory 13, 307-315 (1990; Zbl 0718.47015)]. For any bounded linear operator \(A\), the numerical range of \(A\) is the set \(W(A)= \{\langle Ax,x \rangle:x\in H,\|x\|=1\}\). The author proves that \(\overline {W(\widetilde A)}\subseteq \overline{W(A)}\) for any bounded linear operator \(A\). This result partially supports a conjecture of \textit{I. B. Jung}, \textit{E. Ko} and \textit{C. Percy} [Integral Equ. Oper. Theory 37, 437-448 (2000; Zbl 0996.47008)].
scientific article; zbMATH DE number 1864785