On the $ p $-arithmetic-mean norm of operator pairs and its applications

Feryal Aladsani; Asmahan Alajyan; Silvestru Sever Dragomir; Kais Feki; Feryal Aladsani; Asmahan Alajyan; Silvestru Sever Dragomir; Kais Feki

doi:10.3934/math.2026181

AIMS Mathematics

2026, Volume 11, Issue 2: 4522-4538. doi: 10.3934/math.2026181

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On the $ p $-arithmetic-mean norm of operator pairs and its applications

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Al Ahsa, Saudi Arabia
2.
Applied Mathematics Research Group, ISILC, Victoria University, P.O. Box 14428, Melbourne City, MC 8001, Victoria, Australia
3.
Department of Mathematical and Physical Sciences, La Trobe University, Plenty Road, Bundoora, Melbourne VIC 3086, Australia
4.
Department of Mathematics, College of Science and Arts, Najran University, Najran 66462, Saudi Arabia
5.
University of Sfax, Laboratory Physics-Mathematics and Applications (LR/13/ES-22), Faculty of Sciences of Sfax, Soukra Road Km 3.5, B. P. 1171, 3000, Sfax, Tunisia

Received: 22 September 2025 Revised: 18 January 2026 Accepted: 30 January 2026 Published: 13 February 2026
MSC : 26D15, 47A30, 47A63

The main purpose of this paper is to introduce and study the so-called $ p $-arithmetic-mean norm for pairs of bounded linear operators on a complex Hilbert space $ H $. Specifically, for a pair $ (A, B) $ of bounded linear operators on $ H $, with $ \nu \in [0, 1] $ and $ p > 0 $, we define the following:
$ \|(A, B)\|_{p, \nu} : = \sup\limits_{\|x\| = 1} \Big( (1-\nu)\|Ax\|^p + \nu \|Bx\|^p \Big)^{1/p}. $
We establish, among other results, that for $ \nu \in (0, 1] $ and $ p \in (0, 1] $,
$ \|(A, B)\|_{2p, \nu}^{2p} \leq R \|A-B\|^{2p} + \min \Big\{ \|A\|^{2(1-\nu)p}\|B\|^{2\nu p}, \, \|(1-\nu)|A|^{2}+\nu |B|^{2}\|^p \Big\}. $
Applications are given to off-diagonal operator matrices, and to the particular cases $ (A, B) = (T, T^*) $ and $ (A, B) = \Big(\operatorname{Re}(T), \operatorname{Im}(T)\Big) $, where $ T $ is a bounded linear operator on $ H $, $ T^* $ denotes its adjoint, $ \operatorname{Re}T = \tfrac12(T+T^*) $ is its real part, and $ \operatorname{Im}T = \tfrac{1}{2i}(T-T^*) $ is its imaginary part.
- bounded linear operators,
- Hilbert spaces,
- operator norm,
- arithmetic-mean norm,
- numerical radius,
- Young-type inequalities
Citation: Feryal Aladsani, Asmahan Alajyan, Silvestru Sever Dragomir, Kais Feki. On the $ p $-arithmetic-mean norm of operator pairs and its applications[J]. AIMS Mathematics, 2026, 11(2): 4522-4538. doi: 10.3934/math.2026181

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Abstract

The main purpose of this paper is to introduce and study the so-called $ p $-arithmetic-mean norm for pairs of bounded linear operators on a complex Hilbert space $ H $. Specifically, for a pair $ (A, B) $ of bounded linear operators on $ H $, with $ \nu \in [0, 1] $ and $ p > 0 $, we define the following:

$ \|(A, B)\|_{p, \nu} : = \sup\limits_{\|x\| = 1} \Big( (1-\nu)\|Ax\|^p + \nu \|Bx\|^p \Big)^{1/p}. $

We establish, among other results, that for $ \nu \in (0, 1] $ and $ p \in (0, 1] $,

$ \|(A, B)\|_{2p, \nu}^{2p} \leq R \|A-B\|^{2p} + \min \Big\{ \|A\|^{2(1-\nu)p}\|B\|^{2\nu p}, \, \|(1-\nu)|A|^{2}+\nu |B|^{2}\|^p \Big\}. $

Applications are given to off-diagonal operator matrices, and to the particular cases $ (A, B) = (T, T^*) $ and $ (A, B) = \Big(\operatorname{Re}(T), \operatorname{Im}(T)\Big) $, where $ T $ is a bounded linear operator on $ H $, $ T^* $ denotes its adjoint, $ \operatorname{Re}T = \tfrac12(T+T^*) $ is its real part, and $ \operatorname{Im}T = \tfrac{1}{2i}(T-T^*) $ is its imaginary part.

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