Jointly $ A $-hyponormal $ m $-tuple of commuting operators and related results

Salma Aljawi; Kais Feki; Hranislav Stanković; Salma Aljawi; Kais Feki; Hranislav Stanković

doi:10.3934/math.20241464

AIMS Mathematics

2024, Volume 9, Issue 11: 30348-30363. doi: 10.3934/math.20241464

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Jointly $ A $-hyponormal $ m $-tuple of commuting operators and related results

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, College of Science and Arts, Najran University, Najran 66462, Kingdom of Saudi Arabia
3.
Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, Niš, Serbia

Received: 20 August 2024 Revised: 03 October 2024 Accepted: 10 October 2024 Published: 25 October 2024
MSC : 47A10, 47A12, 47A13, 47B20, 47B37

In this paper, we aim to investigate the class of jointly hyponormal operators related to a positive operator $ A $ on a complex Hilbert space $ \mathcal{X} $, which is called jointly $ A $-hyponormal. This notion was first introduced by Guesba et al. in [Linear and Multilinear Algebra, 69(15), 2888–2907] for $ m $-tuples of operators that admit adjoint operators with respect to $ A $. Mainly, we prove that if $ \mathbf{B} = (B_1, \cdots, B_m) $ is a jointly $ A $-hyponormal $ m $-tuple of commuting operators, then $ \mathbf{B} $ is jointly $ A $-normaloid. This result allows us to establish, for a particular case when $ A $ is the identity operator, a sharp bound for the distance between two jointly hyponormal $ m $-tuples of operators, expressed in terms of the difference between their Taylor spectra. We also aim to introduce and investigate the class of spherically $ A $-$ p $-hyponormal operators with $ 0 < p < 1 $. Additionally, we study the tensor product of specific classes of multivariable operators in semi-Hilbert spaces.
- semi-Hilbert spaces,
- jointly $ A $-hyponormal operators,
- joinlty $ A $-normaloid operators,
- Taylor spectrum,
- spherically $ A $-$ p $-hyponormal,
- tensor product
Citation: Salma Aljawi, Kais Feki, Hranislav Stanković. Jointly $ A $-hyponormal $ m $-tuple of commuting operators and related results[J]. AIMS Mathematics, 2024, 9(11): 30348-30363. doi: 10.3934/math.20241464

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Abstract

In this paper, we aim to investigate the class of jointly hyponormal operators related to a positive operator $ A $ on a complex Hilbert space $ \mathcal{X} $, which is called jointly $ A $-hyponormal. This notion was first introduced by Guesba et al. in [Linear and Multilinear Algebra, 69(15), 2888–2907] for $ m $-tuples of operators that admit adjoint operators with respect to $ A $. Mainly, we prove that if $ \mathbf{B} = (B_1, \cdots, B_m) $ is a jointly $ A $-hyponormal $ m $-tuple of commuting operators, then $ \mathbf{B} $ is jointly $ A $-normaloid. This result allows us to establish, for a particular case when $ A $ is the identity operator, a sharp bound for the distance between two jointly hyponormal $ m $-tuples of operators, expressed in terms of the difference between their Taylor spectra. We also aim to introduce and investigate the class of spherically $ A $-$ p $-hyponormal operators with $ 0 < p < 1 $. Additionally, we study the tensor product of specific classes of multivariable operators in semi-Hilbert spaces.

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