The operator equation Kp=Hδ/2T1/2 (T1/2Hδ+rT1/2)p-δ/δ+r T1/2 Hδ/2 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-1/2 ((TH1/n)-H-1/2+T-r(1/2))T-n(1/2) where n is a natural number. As a continuation, this paper is to consider the equation K-p = (HT1/2)-T-delta/2 ((TH delta)-H-1/2+T-r(1/2))(p-delta/delta+r) T-1/2 H-delta/2 where p > 0, r > 0 and p >= delta > -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. (C) 2007 Elsevier Inc. All rights reserved.