On $ (n_1, \cdots, n_m) $-hyponormal tuples of Hilbert space operators

Sid Ahmed Ould Beinane; Sid Ahmed Ould Ahmed Mahmoud; Sid Ahmed Ould Beinane; Sid Ahmed Ould Ahmed Mahmoud

doi:10.3934/math.20241349

AIMS Mathematics

2024, Volume 9, Issue 10: 27784-27796. doi: 10.3934/math.20241349

Previous Article Next Article

Research article

On $ (n_1, \cdots, n_m) $-hyponormal tuples of Hilbert space operators

Mathematics Department, College of Science, Jouf University, Sakaka, P. O. Box 2014, Saudi Arabia

Received: 18 August 2024 Revised: 13 September 2024 Accepted: 18 September 2024 Published: 26 September 2024
MSC : 47A13

This paper introduces a new class of multivariable operators called $ (n_1, \cdots, n_m) $-hyponormal tuples, which combine joint normal and joint hyponormal operators. A tuple of operators $ \mathcal{Q} = (\mathcal{Q}_1, \; \cdots, \mathcal{Q}_m) $ is said to be an $ (n_1, \cdots, n_m) $-hyponormal tuple for some $ (n_1, \cdots, n_m)\in \mathbb{N}^m $ if
$ \sum\limits_{1\leq k,\;l\leq m}\big\langle[\mathcal{Q}_k^{*n_k}, \;\mathcal{Q}_l^{n_l}]\omega_k\mid \omega_l\big\rangle\geq 0, \quad \forall\; (\omega_k)_{1\leq k\leq m}\in {\mathcal K}^m. $
We show several properties of this class that correspond to the properties of joint hyponormal operators.
- Hilbert space,
- normal operator,
- n-hyponormal operator,
- commuting tuple of operators
Citation: Sid Ahmed Ould Beinane, Sid Ahmed Ould Ahmed Mahmoud. On $ (n_1, \cdots, n_m) $-hyponormal tuples of Hilbert space operators[J]. AIMS Mathematics, 2024, 9(10): 27784-27796. doi: 10.3934/math.20241349

Related Papers:

Abstract

This paper introduces a new class of multivariable operators called $ (n_1, \cdots, n_m) $-hyponormal tuples, which combine joint normal and joint hyponormal operators. A tuple of operators $ \mathcal{Q} = (\mathcal{Q}_1, \; \cdots, \mathcal{Q}_m) $ is said to be an $ (n_1, \cdots, n_m) $-hyponormal tuple for some $ (n_1, \cdots, n_m)\in \mathbb{N}^m $ if

$ \sum\limits_{1\leq k,\;l\leq m}\big\langle[\mathcal{Q}_k^{*n_k}, \;\mathcal{Q}_l^{n_l}]\omega_k\mid \omega_l\big\rangle\geq 0, \quad \forall\; (\omega_k)_{1\leq k\leq m}\in {\mathcal K}^m. $

We show several properties of this class that correspond to the properties of joint hyponormal operators.

References

[1]	A. A. Al-Dohiman, S. A. Ould Ahmed Mahmoud, Class of operators related to $(m, C)$-isometric tuple of commuting operators, J. Inequal. Appl., 2022 (2022), 105. http://doi.org/10.1186/s13660-022-02835-8 doi: 10.1186/s13660-022-02835-8
[2]	N. Ahmad, E. S. Kamel, S. A. Ould Ahmed Mahmoud, $(n_1, \cdots, n_d)$-quasi-$(p, q)$-isometric commuting tuple of operators, Filomat, 37 (2023), 9531–9542. https://doi.org/10.2298/FIL2328531A doi: 10.2298/FIL2328531A
[3]	H. O. Alshamari, Higher order $(n, m)$-Drazin normal operators, J. Inequal. Appl., 2024 (2024), 18. https://doi.org/10.1186/s13660-024-03095-4 doi: 10.1186/s13660-024-03095-4
[4]	A. Athavale, On joint hyponormality of operators, Proc. Amer. Math. Soc., 103 (1988), 417–423. https://doi.org/10.2307/2047154 doi: 10.2307/2047154
[5]	[0.1080/03081087.2019.1593925] H. Baklouti, K. Feki, S. A. Ould Ahmed Mahmoud, Joint normality of operators in semi-Hilbertian spaces, Linear Multilinear A., 68 (2019), 845–866. https://doi.org/10.1080/03081087.2019.1593925 doi: 10.1080/03081087.2019.1593925
[6]	H. Baklouti K. Feki, S. A. Ould Ahmed Mahmoud, Joint numerical ranges of operators in semi-Hilbertian spaces, Linear Algebra Appl., 555 (2018), 266–284. https://doi.org/10.1016/j.laa.2018.06.021 doi: 10.1016/j.laa.2018.06.021
[7]	N. L. Braha, M. Lohaj, F. H. Marevci, S. Lohaj, Some properties of paranormal and hyponormal operators, Bull. Math. Anal. Appl., 1 (2009), 23–35.
[8]	M. Chō, H. Motoyoshi, B. Nastovska, On the joint spectra of commuting tuples of operators and a conjugation, Functional Analysis, Approximation and Computation, 9 (2017), 21–26.
[9]	M. Chō, S. A. Ould Ahmed Mahmoud, $(A, m)$-Symmetric commuting tuple of operators on a Hilbert space, J. Inequal. Appl., 22 (2019), 931–947. https://doi.org/10.7153/mia-2019-22-63 doi: 10.7153/mia-2019-22-63
[10]	J. B. Conway, A course in functional analysis, 2 Eds., New York: Springer, 2007. https://doi.org/10.1007/978-1-4757-4383-8
[11]	B. Fuglede, A commutativity theorem for normal operators, P. Mat. Acad. Sci. USA, 36 (1950), 35–40.
[12]	J. Gleason, S. Richter, $m$-Isometric commuting tuples of operators on a Hilbert space, Integr. Equ. Oper. Theory, 56 (2006), 181–196. https://doi.org/10.1007/s00020-006-1424-6 doi: 10.1007/s00020-006-1424-6
[13]	Y. M. Han, J. H. Son, On quasi-$M$-hyponormal operatros, Filomat, 25 (2011), 37–52. https://doi.org/10.2298/FIL1101037H doi: 10.2298/FIL1101037H
[14]	E. Ko, Properties of a kth root of a hyponormal operator, B. Korean Math. Soc., 40 (2003), 685–692. https://doi.org/10.4134/BKMS.2003.40.4.685 doi: 10.4134/BKMS.2003.40.4.685
[15]	E. Ko, Kth roots of $p$-hyponormal operators are subscalar operators of order 4k, Integr. Equ. Oper. Theory, 59 (2007), 173–187. https://doi.org/10.1007/s00020-007-1519-8 doi: 10.1007/s00020-007-1519-8
[16]	S. A. Ould Ahmed Mahmoud, M. Chō, J. E. Lee, On $(m, C)$-isometric commuting tuples of operators on a Hilbert space, Results Math., 73 (2018), 51. https://doi.org/10.1007/s00025-018-0810-0 doi: 10.1007/s00025-018-0810-0
[17]	S. A. Ould Ahmed Mahmoud, H. O. Alshammari, Joint $m$-quasihyponormal on a Hilbert space, Ann. Funct. Anal., 12 (2021), 42. https://doi.org/10.1007/s43034-021-00130-z doi: 10.1007/s43034-021-00130-z
[18]	S. Mecheri, On the normality of operators, Revista Colombiana de Matemáticas, 39 (2005), 87–95.
[19]	M. Guesba, E. M. Ould Beiba, S. A. Ould Ahmed Mahmoud, Joint $A$-hyponormality of operators in semi-Hilbert spaces, Linear Multilinear A., 69 (2021), 2888–2907. https://doi.org/10.1080/03081087.2019.1698509 doi: 10.1080/03081087.2019.1698509
[20]	C. R. Putnam, On normal operators in Hilbert space, Am. J. Math., 73 (1951), 357–362. https://doi.org/10.2307/2372180 doi: 10.2307/2372180
[21]	S. G. Shi, Z. W. Fu, S. Z. Lu, On the compactness of commutators of Hardy operators, Pac. J. Math., 307 (2020), 239–256. https://doi.org/10.2140/pjm.2020.307.239 doi: 10.2140/pjm.2020.307.239
[22]	J. G. Stamofli, Hyponormal operatros, Pac. J. Math., 12 (1962), 1453–1458.
[23]	L. Zhang, S. G. Shi, A characterization of central BMO space via the commutator of fractional Hardy operator, J. Funct. Space., 2020 (2020), 3543165. https://doi.org/10.1155/2020/3543165 doi: 10.1155/2020/3543165
[24]	F. Zuo, H. L. Zuo, Spectral properties and characterization of quasi-$n$-hyponormal operators, J. Math. Inequal., 16 (2022), 965–974. https://doi.org/10.7153/jmi-2022-16-65 doi: 10.7153/jmi-2022-16-65

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)