Some properties of the generalized Aluthge transform
If \(T\) is a bounded operator on a complex separable Hilbert space \(\mathcal H\), the Aluthge transform is defined by \(\widetilde {T} = | T| ^{\frac{1}{2}} U | T| ^{\frac{1}{2}}\) where \(T = UT\) is the polar decomposition of \(T\). The generalized Aluthge transform is defined by \(\widetilde {T}(t) = | T| ^{t} U | T| ^{1-t}\) for \(0< t<1\), \(T(0) = U^*UU| T| \), and \(T(1) = | T| U\). The authors' main result is that if \(\rho(A) = \{\lambda \in \mathbb C : R(A-\lambda I)\) is closed\(\} \setminus \{0 \} \), then \(\rho(T) = \rho(\widetilde {T}(t))\) for \(0\leq t \leq 1\). Some additional results on the numerical range of the generalized Aluthge transform are also obtained.
scientific article; zbMATH DE number 2157946