On weakly unitarily invariant norm and the Aluthge transformation (Q1406303)

The main result: \(|||f(P^\lambda UP^{1-\lambda})|||\leq \max\{|||f(T)|||, |||U^* f(T)U+ f(0)(I- U^* U)|||\}\), where \(T\in B(H)\) is a bounded linear operator on a Hilbert space \(H\), \(f\) is a polynomial, and \(|||\cdot|||\) is a seminorm on \(H\) which satisfies the following two conditions: a) \(\exists\gamma> 0\) such that \(|||X|||\leq \gamma\|X\|\) \(\forall X\in B(H)\), and b) \(\|||S^*XS|||\leq \|S\|^2|||X|||\) \(\forall X\), \(S\in B(H)\); here \(\|\cdot\|\) is the Hilbert operator norm. As a consequence of this fact it is proved that for some seminorms (including the \(\rho\)-radii, \(0< \rho\leq 2\)), \(|||f(P^\lambda UP^{1-\lambda})|||\leq |||f(T)|||\).