BISHOP’S PROPERTY(β) AND WEIGHTED CONDITIONAL TYPE OPERATORS INk-QUASI CLASS A∗

Özet

An operatorTis said to bek-quasi classA∗noperator ifT∗k(|Tn+1|2n+1−|T∗|2)Tk≥0,for some positive integersnandk. In this paper,we prove that thek-quasi classA∗noperators have Bishop,s property (β). Then, we givea necessary and sufficient condition forT⊗Sto be ak-quasi classA∗noperator, wheneverTandSare both non-zero operators. Moreover, it is shown that the Riesz idempotentfor a non-zero isolated pointλ0of ak-quasi classA∗noperatorTsayRi, is self-adjointandran(Ri) =ker(T−λ0) =ker(T−λ0)∗. Finally, as an application in the last section,a necessary and sufficient condition is given in such a way that the weighted conditionaltype operators onL2(Σ), defined byTw,u(f) :=wE(uf), belong tok-quasi-A∗n class.