Abstract:

Given a bounded linear operator T with canonical polar decomposition T=V|T|, the Aluthge transform of T is the operator $Δ(T):=sqrt{|T|} V sqrt{|T|}. For P an arbitrary positive operator such that V P=T, we define the extended Aluthge transform of T associated with P by Δ_{P}(T):=sqrt{P} V sqrt{P}.

First, we establish some basic properties of Δ_{P}; 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 Δ_{P} leaves invariant the class of complex symmetric operators. We also study how Δ_{P} 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 x 2 operator matrix built from the pair. Thus, the theory of extended Aluthge transforms yields results for spherical Aluthge transforms.