On weakly unitarily invariant norm and the \(\lambda\)-Aluthge transformation for invertible operator
For any bounded linear operator \(T\) on a Hilbert space and \(0\leq\lambda\leq 1\), the \(\lambda\)-Aluthge transform \(T_\lambda\) of \(T\) is by definition the operator \(P^\lambda UP^{1-\lambda}\), where \(U\) and \(P\) are the operators in the polar decomposition \(T= UP\) of \(T\), with \(P= (T*T)^{1/2}\) and \(U\) a partial isometry satisfying \(U*UP= P\). In this paper, the author first proves that if \(T\) acts on a space \(H\) of dimension 2, then \(\| T_\lambda\|=\| T_{1- \lambda}\|\) for any \(0\leq \lambda\leq 1\), where \(\|\cdot\|\) denotes the spectral norm. By way of an example, he also shows that this is no longer the case if \(\dim H\geq 3\). Next, he proves that if \(T\) is invertible and \(B\) commutes with \(T\), then \(|||P^\lambda BUP^{1-\lambda}|||\leq |||BT|||\) for any weakly unitarily invariant norm \(|||\cdot|||(|||W*XW|||= |||X|||\) for any operators \(X\) and \(U\) with \(U\) unitary). This generalizes a previous result by the same author that \(|||p(T_\lambda)|||\leq |||p(T)|||\) for any polynomial \(p\).
scientific article; zbMATH DE number 5077726