Square of \(w\)-hyponormal operators (Q5937209)

A bounded linear operator \(T\) on a complex Hilbert space with polar decomposition \(T= U|T|\) is said to be \(w\)-hyponormal if \(||T|^{{1\over 2}} U|T|^{{1\over 2}}|\geq |T|\geq ||T|^{{1\over 2}} U^*|T|^{{1\over 2}}\). It is shown that the square of a \(w\)-hyponormal operator itself is \(w\)-hyponormal. This generalizes a result of Althuge and Wang who for this implication additionally needed the assumption of invertibility for \(T\).