The Extended Aluthge Transform

Given a bounded linear operator T with canonical polar decomposition \(T \equiv V\left |T\right |\), the Aluthge transform of T is the operator \(\Delta (T):=\sqrt {\left |T\right |} V \sqrt {\left |T\right |}\). For P an arbitrary positive operator such that V P = T, we define the extended Aluthge transform of T associated with P by \(\Delta _P(T):=\sqrt {P} V \sqrt {P}\). First, we establish some basic properties of ΔP; second, we study the fixed points of the extended Aluthge transform; third, we consider the case when T is an idempotent; next, we discuss whether ΔP leaves invariant the class of complex symmetric operators. We also study how ΔP transforms the numerical radius and numerical range. As a key application, we prove that the spherical Aluthge transform of a commuting pair of operators corresponds to the extended Aluthge transform of a 2 × 2 operator matrix built from the pair; thus, the theory of extended Aluthge transforms yields results for spherical Aluthge transforms.