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

Year 2021, Volume: 18 Issue: 2, 101 - 116, 01.11.2021

Djilali Bekai Benali Abdelkader , Hakem Alı

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.

Keywords

A-positive , A-isometry , Semi-Hilbertian space , A-positive , A-isometry , A-normal , A-quasi-normal

Thanks

The authors would like to express their gratitude to the referee. We are very grateful for his help, his careful observations, and his careful reading, which led to the improvement of the article.

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