On the polar decomposition of the Duggal transformation and related results

Let T = U |T | be the polar decomposition of a bounded operator T on a Hilbert space. The transformation Δ(T ) = |T |1/2U |T |1/2 is called the Aluthge transformation, and Γ(T ) = |T |U is called the Duggal transformation of T. We discuss Aluthge transformation and Duggal transformation of binormal operators and centered operators. We obtain results about the polar decomposition of Duggal transformation. We give necessary and sufficient conditions for Γ(T ) to have the polar decomposition Γ(T ) = Γ(U) |Γ(T )|. As a consequence we get Γ(T ) = Γ(U) |Γ(T )| to be the polar decomposition of Γ(T ) if T is binormal. Mathematics subject classification (2000): 47A05, 47B20.