SOME NEW CHARACTERIZATIONS OF PARTIAL ISOMETRIES IN RINGS WITH INVOLUTION

Abstract

In this paper, some characterizations of partial isometries, normal elements and strongly $EP$ elements are given by the construction of $EP$ elements. At the same time, the partial isometry elements are characterized by the existence of solutions of equations in rings in a given set, and also by the general form of solutions of given equations.

Keywords

• Group inverse
• MP-invertible element
• partial isometry element
• $EP$ element
• strongly $EP$ element
• normal element

References

1. A. Ben-Israel and T. N. E Greville, Generalized Inverses: Theory and Applications, 2nd. ed., Springer, New York, 2003.
2. S. Karanasios, EP elements in rings and semigroup with involution and $C^*$-algebras, Serdica Math. J., 41 (2015), 83-116.
3. J. J. Koliha and P. Patricio, Elements of rings with equal spectral idempotents, J. Aust. Math. Soc., 72 (2002), 137-152.
4. D. Mosic and D. S. Djordjevic, Partial isometries and EP elements in rings with involution, Electron. J. Linear Algebra, 18 (2009), 761-772.
5. D. Mosic and D. S. Djordjevic, Further results on partial isometries and EP elements in rings with involution, Math. Comput. Modelling, 54 (2011), 460- 465.
6. D. Mosic, D. S. Djordjevic and J. J. Koliha, EP elements in rings, Linear Algebra Appl., 431 (2009), 527-535.
7. Y. C. Qu, J. C. Wei and H. Yao, Characterizations of normal elements in rings with involution, Acta. Math. Hungar., 156(2) (2018), 459-464.
8. Z. C. Xu, R. J. Tang and J. C. Wei, Strongly EP elements in a ring with involution, Filomat, 34(6) (2020), 2101-2107.

9. S. Z. Xu, J. L. Chen and J. Benítez, EP elements in rings with involution, Bull. Malays. Math. Sci. Soc., 42 (2019), 3409-3426.
10. R. J. Zhao, H. Yao and J. C. Wei, Characterizations of partial isometries and two special kinds of EP elements, Czechoslovak Math. J., 70(145) (2020), 539- 551.
11. R. J. Zhao, H. Yao and J. C. Wei, EP elements and the solutions of equation in rings with involution, Filomat, 32(13) (2018), 4537-4542.