An investigation of exponentially $m$-isometries within the framework of semi-Hilbert spaces

Abstract

In this paper, the concept of exponentially $m$-isometry [31] on a Hilbert space is generalized when an additional semi-inner product is considered. We present a comprehensive study of the algebraic properties and characterizations of operators within this extended class. Furthermore, we explore the dynamical behavior of these operators and conclude with an analysis of their spectral properties.

Keywords

• m-Isometric operator
• semi-inner product
• exponentially (m
• A)-isometric operator
• N-supercyclicity
• spectrum

Supporting Institution

Dijana Mosić is supported by the Ministry of Science, Technological Development and Innovation, Republic of Serbia, grant number 451-03-137/2025-03/200124, and by the project ”Linear operators: invertibility, spectra and operator equations” supported by the Branch of SANU in Niš, grant no. O-30-22.

Ethical Statement

This research was carried out free of any conflicts of interest or personal affiliations that could potentially affect the integrity of the findings.

References

1. [1] Abeer A. Al Dohiman, M. A. Aouichaoui and O. A. M. Sid Ahmed, Structure and applications of n-quasi exponentially m-isometric operators, Adv. Oper. Theory 10 (3), 1-16, 2025.
2. [2] Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory Vol. 50, American Mathematical Society, Providence, RI, 2002.
3. [3] P. Aiena and M. T. Biondi, Some spectral mapping theorems through local spectral theory, Rend. Circ. Mat. Palermo 53, 165-184, 2004.
4. [4] J. Agler, A disconjugacy theorem for Toeplitz operators, American Journal of Mathematics 112 (1), 1-14, 1990.
5. [5] J. Agler and M. Stankus, m-isometric transformations of Hilbert space, I, Integral Equ. Oper. Theory 21 (4), 383, 1995.
6. [6] J. Agler and M. Stankus, m-isometric transformations of Hilbert space, II, Integral Equ. Oper. Theory 23 (1), 1-48, 1995.
7. [7] J. Agler and M. Stankus, m-isometric transformations of Hilbert space, III, Integral Equ. Oper. Theory 24 (4), 379, 1996.
8. [8] S. I. Ansari and P. S. Bourdon, Some properties of cyclic operators, Acta Sci. Math. 63 (1-2), 195-207, 1997.

9. [9] M. A. Aouichaoui, A note on the spectral mapping theorem for point spectrum, Rend. Circ. Mat. Palermo Series 2 73 (7), 2869-2875, 2024.
10. [10] M. A. Aouichaoui, A note on partial-A-isometries and some applications, Quaest. Math. 47 (3), 515-535, 2024.
11. [11] M. A. Aouichaoui, $(m, \mathcal{N}_{A})$-isometries and their harmonious applications to Hilbert- Schmidt operators, Linear Algebra Appl. 697, 309-331, 2024.
12. [12] M. L. Arias, G. Corach and M. Celeste Gonzalez, Partial isometries in semi-Hilbertian spaces, Linear Algebra Appl. 428 (7), 1460-1475, 2008.
13. [13] M. L. Arias, G. Corach and M. C. Gonzalez, Metric properties of projections in semi- Hilbertian spaces, Integral Equ. Oper. Theory 62 (1), 11-28, 2008.
14. [14] F. Bayart, m-isometries on Banach spaces, Math. Nachr. 284, 2141-2147, 2011.
15. [15] F. Bayart, É. Matheron, Dynamics of linear operators, Vol. 179, Cambridge University Press, 2009.
16. [16] T. Bermúdez, A. Martinón and J. Agustín Noda, An isometry plus a nilpotent operator is an m-isometry. Applications, J. Math. Anal. Appl. 407 (2), 505-512, 2013.
17. [17] T. Bermúdez, A. Saddi and H. Zaway, (A,m)-isometries on Hilbert spaces, Linear Algebra Appl. 540, 95-111, 2018.
18. [18] T. Bermúdez and I. Marrero, Antonio Martinón, On the Orbit of an m-Isometry, Integral Equ. Oper. Theory 64, 487-494, 2009.
19. [19] T. Bermúdez, A. Martinón and V. Müller, (m, q)-isometries on metric spaces, J. Operator Theory 72 (2), 313-329, 2014.
20. [20] M. Chô, S. Ôta and K. Tanahashi, Invertible weighted shift operators which are misometries, Proc. Amer. Math. Soc. 141, 4241-4247, 2013.
21. [21] R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proceedings of the American Mathematical Society 17 (2), 413-415, 1966.
22. [22] M. Faghih-Ahmadi, Powers of A-m-isometric operators and their supercyclicity, Bull. Malays. Math. Sci. Soc. 39 (3), 901-911, 2016.
23. [23] M. Faghih Ahmadi and K. Hedayatian, Hypercyclicity and supercyclicity of misometric operators, Rocky Mountain J. Math. 42 (1), 15-23, 2012.
24. [24] N. S. Feldman, N-supercyclic operators, Studia Math 151 (2), 141-159, 2002.
25. [25] K. G. Grosse-Erdmann and A. Peris Manguillot, Linear chaos, Springer Science & Business Media, 2011.
26. [26] C. Gu, The (m; q)-isometric weighted shifts on $l_p$ spaces, Integral Equ. Oper. Theory 82, 157-187, 2015.
27. [27] H. M. Hilden and L. J.Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J. 23, 110-115, 1974.
28. [28] I. Marrero, On the Symbols of Strictly m-Null Elementary Operators, Mathematics 13 (12), 2026, 2025.
29. [29] A. Mohammadi-Moghaddam and K. Hedayatian, On the dynamics of the d-tuples of m-isometries, Rocky Mt. J. Math, 283-305, 2019.
30. [30] M. Z. Nashed, Inner, outer, and generalized inverses in Banach and Hilbert spaces, Numer. Funct. Anal. Optim. 9 (3-4), 261-325, 1987.
31. [31] M. Salehi and K. Hedayatian, Exponentially m-isometric operators on Hilbert spaces, J. Math. Anal. Appl. 522 (1), 127015, 2023.
32. [32] O. A. M. Sid Ahmed and M. A. Aouichaoui, Advancements in the generalization of m-isometries: introducing the q-partial-(m,A)-isometry class, Linear Multilinear Algebra 73.6 (6), 1098-1120, 2025.
33. [33] O. A. M. Sid Ahmed and A. Saddi, A-m-Isometric operators in semi-Hilbertian spaces, Linear Algebra Appl. 436 (10), 3930-3942, 2012.