ON THE PARTIAL NORMALITY OF A CLASS OF BOUNDED OPERATORS

Abstract

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