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.

Keywords

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

References

1. [1] J.B. Conway, A Course in Functional Analysis, Springer Verlag, Berlin -Heildelberg-New York, 1990.
2. [2] C. R. Putnam, “On normal operators in Hilbert space,” American Journal of Mathematics, vol. 73, pp. 357-362, 1951.
3. [3] O. A. Mahmoud Sid Ahmed and O. B. Sid Ahmed, “On the Classes of (𝑛, 𝑚) power-𝐷-Normal and(𝑛, 𝑚)power-𝐷- Quasinormal,” Operators And Matrices, vol. 13, no. 3, pp. 705-73, 2019.
4. [4] M. L. Arias, G. Corach, and M. C. Gonzalez, “Partial isometries in semi- Hilbertian spaces,” Linear Algebra and its Applications, vol. 428, no. 7, pp. 1460-1475, 2008.
5. [5] M. L. Arias, G. Corach, M. C. Gonzalez, “Metric properties of projections in semi- Hilbertian spaces,” Integral Equations Operator Theory , vol. 62, no. 1, pp. 11-28, 2008.
6. [6] M. L. Arias, G. Corach, and M. C. Gonzalez, “Lifting properties in operator ranges,” Acta Scientiarum Mathematicarum (Szeged), vol. 75, no. (3-4), pp. 635-653, 2009.
7. [7] O. A. Mahmoud Sid Ahmed and A. Saddi, “A-m-Isomertic operators in semi-Hilbertian spaces,” Linear Algebra and its Applications, vol. 436, pp. 3930-3942, 2012.
8. [8] Ould Ahmed Mahmoud Sid Ahmed and Abdelkader Benali, “Hyponormal And 𝑘-Quasi-hyponormal Operators On Semi-Hilbertian Spaces,” The Australian Journal of Mathematical Analysis and Applications, vol. 13, no. 1, pp. 1- 22, 2016.

9. [9] A. Saddi, “ A-Normal operators in Semi-Hilbertian spaces,” The Australian Journal of Mathematical Analysis and Applications, vol. 9, no. 1, pp. 1-12, 2012.
10. [10] S. H. Jah, “Class Of (𝐴, 𝑛) power Quasi-normal Opertors in Semi Hilbertian Space,” Internationl Journal of Pure and Applied Mathematics, vol. 93, no. 1, pp. 61-83, 2014.
11. [11] R. G. Douglas, “On majorization, factorization and range inclusion of operators in Hilbert space,” Proceedings of the American Mathematical Society, vol. 17, pp. 413-416, 1966.
12. [12] S. R. Caradus, “Operator Theory of the Generalized Inverse,” Queens Papers in Pure and Applied Math, vol. 38, 2004.
13. [13] C. F. King, “A note of Drazin inverses,” Pacific Journal of Mathematics, vol. 70, no. 2, pp. 383–390, 1977.
14. [14] S. L. Campbell and C. D. Meyer, “Generalized Inverses of Linear Transformations,” Society for Industrial and Applied Mathematics, 2009.
15. [15] D.S. Djordjevic and V. Rakocevic, “Lectures on Generalized Inverse,” Faculty of Science and Mathematics, University of Nice, 2008.
16. [16] M. Dana and R. Yousfi, “On the classes of 𝐷-normal operators and 𝐷-quasi-normal operators,” Operators and Matrices, vol. 12, no. 2, pp. 465–487, 2018.
17. [17] G. Wang, Y. Wei, and S. Qiao, “Generalized Inverses: Theory and Computations,” Graduate Series in Mathematics, vol. 5, Beijing, 2004.
18. [18] A. A. S. Jibril, “On-power Normal Operators,” The Journal for Science and Engineering, vol. 33, no. 2A, pp. 247- 251, 2008.
19. [19] O. A. M. Sid Ahmed, “On the class of n-power quasi-normal operators on Hilbert spaces,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 2, pp. 213–228, 2011.