Extended Aluthge Transforms and Applications (Research on structure of operators by order and related topics)

Abstract

Given a bounded linear operator T with canonical polar decomposition T = V|T|, the Aluthge transform of T is the operator Δ(T) := √|T|V √|T|. For P an arbitrary positive operator such that VP = T, we define the extended Aluthge transform of T associated with P by Δp(T) := √PV √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 x 2 operator matrix built from the pair; thus, the theory of extended Aluthge transforms yields results for spherical Aluthge transforms.