Polynomially partial isometric operators

Abstract

In order to extend the notion of semi-generalized partial isometries and partial isometries, we introduce a new class of operators called polynomially partial isometries. Since this new class of operators contains semi-generalized partial isometries, partial isometries, isometries and co-isometries, we proposed a wider class of operators. Several basic properties of polynomially partial isometries and some invariant subspaces of corresponding operators are presented. We study decomposition theorems and spectral theorems for polynomially partial isometries, generalizing some well-known results for partial isometries and semi-generalized partial isometries to polynomially partial isometries. Applying polynomially partial isometries, we solve some equations.

Keywords

• partial isometry
• isometry
• decomposition theorem
• Hilbert space

References

1. [1] A. Alahmari, M. Mabrouk and M.A. Taoudi, Discussions on partial isometries in Banach spaces and Banach algebras, Bull. Korean Math. Soc. 54 (2), 485-495, 2017.
2. [2] S.A. Aluzuraiqi and A.B. Patel, On n-normal operators, General Math. Notes 1, 61-73, 2010.
3. [3] C. Apostol, Propriétés de certains operateurs bornés des espaces de Hilbert II, Rev. Roum. Math. Purs Appl. 12, 759-762, 1967.
4. [4] M.L. Arias and M. Mbekhta, On partial isometries in C*–algebras, Studia Math. 205 (1), 71-82, 2011.
5. [5] C. Badea and M. Mbekhta, Operators similar to partial isometries, Acta Sci. Math. (Szeged) 71, 663-680, 2005.
6. [6] S.K. Berberian, Approximate proper vectors, Proc. Amer. Math. Soc. 13, 111-114, 1962.
7. [7] M. Chō, J.E. Lee, K. Tanahashi and A. Uchiyama, Remarks on n-normal operators, Filomat 32 (15), 5441-5451, 2018.
8. [8] M. Chō and B. Načevska Nastovska, Spectral properties of n-normal operators, Filomat 32 (14), 5063-5069, 2018.

9. [9] D.S. Djordjević, M. Chō and D. Mosić, Polynomially normal operators, Ann. Funct. Anal. 11, 493-504, 2020.
10. [10] B.P. Duggal and I.H. Kim, On nth roots of normal operators, Filomat 34 (8), 2797- 2803, 2020.
11. [11] I. Erdelyi and F.R. Miller, Decomposition theorems for partial isometries, J. Math. Anal. Appl. 30, 665-679, 1970.
12. [12] F.J. Fernández-Polo and A. Peralta, Partial isometries: a survey, Adv. Oper. Theory 3 (1), 75-116, 2018.
13. [13] Z. Garbouj and H. Skhiri, Semi-generalized partial isometries operators, Results Math. 75, 15, 2020.
14. [14] P.R. Halmos and J.E. McLaughlin, Partial isometries, Pac. J. Appl. Math. 13, 585- 596, 1963.
15. [15] P.R. Halmos and L.J. Wallen, Powers of partial isometries, J. Math. Mech. 19, 657- 663, 1969/1970.
16. [16] O.A. Mahmoud Sid Ahmed, Generalization of m-partial isometries on a Hilbert space, Int. J. Pure Appl. Math. 104 (4), 599-619, 2015.
17. [17] M. Mbekhta, Partial isometries and generalized inverses, Acta Sci. Math. (Szeged) 70, 767-781, 2004.
18. [18] D. Mosić and D.S. Djordjević, Partial isometries and EP elements in Banach algebras, Abstr. Appl. Anal. 2011, Article ID 540212, 9 pages, 2011.
19. [19] D. Mosić and D.S. Djordjević, Weighted partial isometries and weighted–EP elements in C*-algebra, Appl. Math. Comput. 265, 17-30, 2015.
20. [20] A. Saddi and O.A. Mahmoud Sid Ahmed, m-partial isometries on Hilbert spaces, Internat. J. Functional Analysis, Operator Theory and Applications 2 (1), 67-83, 2010.
21. [21] C. Schmoeger, Partial isometries on Banach spaces, Seminar LV, No. 20, 13 pp. 28.02.2005.