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This paper discusses some spectral properties of class operators for , , , and . It is shown that if is a class operator, then the Riesz idempotent of with respect to each nonzero isolated point spectrum is selfadjoint and . Afterwards, we prove that every class operator has SVEP and property , and Weyl's theorem holds for when .

Firstly, we will show the following extension of the results on powers of -hyponormal and -hyponormal operators: let and be positive integers, if is -hyponormal for , then: (i) in case , and hold, (ii) in case , and hold. Secondly, we will show an estimation on powers of -hyponormal operators for which implies the best possibility of our results. Lastly, we will show a parallel estimation on powers of -hyponormal operators as follows: let , then the following hold for each positive integer and : (i) there exists a log-hyponormal operator such that , (ii) there exists a -hyponormal operator such that .

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=T12(T12H1nT12)nT12 where n is a natural number. As a continuation, this paper is to consider the equation Kp=Hδ2T12(T12Hδ+rT12)p−δδ+rT12Hδ2 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.