The operator equation Kp = H δ2 T 12 (T 12 Hδ+rT 12 ) p−δ δ+r T 12 H δ2 and its applications

Let H and K be bounded positive operators on a Hilbert space, and assume that H is nonsingular. Based on Pedersen and Takesaki’s research on the operator equation K = THT , Furuta and Bach gave deep discussion on the equation K = T 12 (T 12 H 1 n T 12 )nT 12 where n is a natural number. As a continuation, this paper is to consider the equation Kp = H δ2 T 12 (T 12 Hδ+rT 12 ) p−δ δ+r T 12 H δ2 where p > 0, r > 0 and p δ > −r. As applications, we prove that the inclusion relations among class wA(p, r) operators are strict and show a generalization of Aluthge’s result. © 2007 Elsevier Inc. All rights reserved