Fuglede-Putnam Type Theorems Via the Moore-Penrose Inverse and Aluthge Transform

Let A, B ∈ B(H), where H is a Hilbert space. Let T and T † denote the Aluthge transform and the Moore-Penrose inverse of T , respectively. We show that (i) if A∗ is quasinormal, then ((A)†, (B ((A)†, (B)†). In general, (T )† = T †. Finally, we give some applications to the Lambert multiplication operator MwEMu on L2(Σ), where E is the conditional expectation operator.