An operator T is said to be k-quasi class A*n operator if T*ᵏ (|Tⁿ⁺¹|²/ⁿ⁺¹− |T*|² ) Tᵏ ≥ 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 L² (Σ), defined by Tw,u(f) := wE(uf), belong to k-quasi- A*n class.Publisher's Versio
