On the partial normality of a class of bounded operators

Y. Estaremi

In this paper, some various partial normality classes of weighted conditional expectation type operators on $L^{2}(\Sigma)$ are investigated. For a weakly hyponormal weighted conditional expectation type operator $M_wEM_u$, we show that the conditional Cauchy-Schwartz inequality for u and w becomes an equality. Assuming this equality, we then show that the joint point spectrum is equal to the point spectrum of $M_wEM_u$. Also, we compute the approximate point spectrum of $M_wEM_u$ and we prove that under a mild condition the approximate point spectrum and the spectrum of $M_wEM_u$ are the same.