BISHOP’S PROPERTY (β) an‎d WEIGHTED CONDITIONAL TYPE OPERATORS IN k-QUASI CLASS A∗ n

An operator T is said to be k-quasi class A ∗ n operator if T ∗k |T n+1| 2 n+1 − |T ∗ | 2 T k ≥ 0, for some positive integers n and k. In this paper, we prove that the k-quasi class A ∗ n operators have Bishop, s property (β). Then, we give a necessary and sufficient condition for T ⊗S to be a k-quasi class A ∗ n operator, whenever T and S are both non-zero operators. Moreover, it is shown that the Riesz idempotent for a non-zero isolated point λ0 of a k-quasi class A ∗ n operator T say Ri, is self-adjoint and ran(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 conditional type operators on L 2 (Σ), defined by Tw,u(f) := wE(uf), belong to k-quasi- A ∗ n class.