The composition of conditional expectation and multiplication operators

摘要

In this paper, we give some necessary and sufficient conditions forweighted conditional type operators on L~2(Σ) to be centered. Consequently, undera weak condition we get that the weighted conditional type operator M_wEM_uis centered if and only if the conditional type H?lder inequality turn into equalityfor w, u. As an important consequence we get that the Aluthge transformationsof a conditional type operator is always centered and it's point spectrum andjoint point spectrum are equal. Also, we obtain that normality, hyponormality,p-hyponormality and centeredness of all operators of the form of EM_u are coincided.In addition we get that all operators of the form M_vM_u?EM_u are alwaysnormal and so centered, when the function v is A-measurable. Moreover, we seethat the point spectrum and joint-point spectrum of weighted conditional typeoperators are equal, when they are centered. As an important consequence, weget that the point spectrum and joint point spectrum of all operators of the formM_vM_u?EM_u on L~2(Σ) are equal when v is A-measurable and S(vE(u|~2)) = X.