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

https://izlik.org/JA98XF76SR

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.

References

• [1] J.B. Conway, A Course in Functional Analysis, Springer Verlag, Berlin -Heildelberg-New York, 1990.
• [2] C. R. Putnam, “On normal operators in Hilbert space,” American Journal of Mathematics, vol. 73, pp. 357-362, 1951.
• [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] 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] 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] 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] 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] 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] 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] 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] 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] S. R. Caradus, “Operator Theory of the Generalized Inverse,” Queens Papers in Pure and Applied Math, vol. 38, 2004.
• [13] C. F. King, “A note of Drazin inverses,” Pacific Journal of Mathematics, vol. 70, no. 2, pp. 383–390, 1977.
• [14] S. L. Campbell and C. D. Meyer, “Generalized Inverses of Linear Transformations,” Society for Industrial and Applied Mathematics, 2009.
• [15] D.S. Djordjevic and V. Rakocevic, “Lectures on Generalized Inverse,” Faculty of Science and Mathematics, University of Nice, 2008.
• [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] G. Wang, Y. Wei, and S. Qiao, “Generalized Inverses: Theory and Computations,” Graduate Series in Mathematics, vol. 5, Beijing, 2004.
• [18] A. A. S. Jibril, “On-power Normal Operators,” The Journal for Science and Engineering, vol. 33, no. 2A, pp. 247- 251, 2008.
• [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.