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 ∗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.

Keywords

• Weighted translation,pre-frame,conditional expectation,measurable function.

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