Bishop’s property (?) and weighted conditional type operators in k-quasi class A*n

Abstract

An operator T is said to be k-quasi class A*n operator if T*? (|T??¹|²/??¹? |T*|² ) T? ? 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 (?). 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 ?0 of a k-quasi class A*n operator T say R?, is self-adjoint and ran(R?) = ker(T ???) = ker(T ???)*. 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 (?), defined by Tw,u(f) := wE(uf), belong to k-quasi- A*n class.