On the Classes of (n, m) Power (D, A)-Normal and (n,m) Power (D, A)-Quasinormal Operators in Semi-Hilbertian Space

Abstract

The concept of (𝑛, 𝑚) power 𝐷-normal operators on Hilbertian space is defined by Ould Ahmed Mahmoud Sid Ahmed and Ould Beinane Sid Ahmed in [1]. In this paper we introduce a new classes of operators on semi-Hilbertian space (ℋ, ∥. ∥𝐴) called (𝑛, 𝑚) power-(𝐷, 𝐴)-normal denoted [(𝑛, 𝑚)𝐷𝑁]𝐴 and (𝑛, 𝑚) power-(𝐷, 𝐴)-quasi-normal denoted [(𝑛, 𝑚)𝐷𝑄𝑁]𝐴 associated with a Drazin invertible operator using its Drazin inverse. Some properties of [(𝑛, 𝑚)𝐷𝑁]𝐴 and [(𝑛, 𝑚)𝐷𝑄𝑁]𝐴 are investigated and some examples are also given. An operator 𝑇 ∈ ℬ𝐴 (ℋ) is said to be (n, m) power-(𝐷, 𝐴)- normal for some positive operator 𝐴 and for some positive integers 𝑛 and 𝑚 if (𝑇𝐷)𝑛(𝑇⋕)𝑚 = (𝑇⋕)𝑚(𝑇𝐷)𝑛.