The operator equation 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 where n is a natural number. As a continuation, this paper is to consider the equation 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.