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16 pages, 329 KiB

Open AccessArticle

On Some Inequalities for the Generalized Euclidean Operator Radius

by Mohammad W. Alomari, Gabriel Bercu, Christophe Chesneau and Hala Alaqad

Axioms 2023, 12(6), 542; https://doi.org/10.3390/axioms12060542 - 31 May 2023

Cited by 1 | Viewed by 1345

In the literature, there are many criteria to generalize the concept of a numerical radius; one of the most recent and interesting generalizations is the so-called generalized Euclidean operator radius, which reads: [...] Read more.

ω_{p} (T_{1}, \dots, T_{n}) : = sup_{∥x∥ = 1} {(\sum_{i = 1}^{n} {|⟨T_{i} x, x⟩|}^{p})}^{1 / p}, p \geq 1,

for all Hilbert space operators

T_{1}, \dots, T_{n}

. Simply put, it is the numerical radius of multivariable operators. This study establishes a number of new inequalities, extensions, and generalizations for this type of numerical radius. More precisely, by utilizing the mixed Schwarz inequality and the extension of Furuta’s inequality, some new refinement inequalities are obtained for the numerical radius of multivariable Hilbert space operators. In the case of

n = 1

, the resulting inequalities could be considered extensions and generalizations of the classical numerical radius. Full article

(This article belongs to the Special Issue Symmetry of Nonlinear Operators)

11 pages, 275 KiB

Open AccessArticle

Improvement of Furuta’s Inequality with Applications to Numerical Radius

by Mohammad W. Alomari, Mojtaba Bakherad, Monire Hajmohamadi, Christophe Chesneau, Víctor Leiva and Carlos Martin-Barreiro

Mathematics 2023, 11(1), 36; https://doi.org/10.3390/math11010036 - 22 Dec 2022

Cited by 4 | Viewed by 1672

Abstract

In diverse branches of mathematics, several inequalities have been studied and applied. In this article, we improve Furuta’s inequality. Subsequently, we apply this improvement to obtain new radius inequalities that not been reported in the current literature. Numerical examples illustrate the main findings. [...] Read more.

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

18 pages, 320 KiB

Open AccessArticle

On the Dragomir Extension of Furuta’s Inequality and Numerical Radius Inequalities

by Mohammad W. Alomari, Gabriel Bercu and Christophe Chesneau

Symmetry 2022, 14(7), 1432; https://doi.org/10.3390/sym14071432 - 12 Jul 2022

Cited by 3 | Viewed by 1719

Abstract

In this work, some numerical radius inequalities based on the recent Dragomir extension of Furuta’s inequality are obtained. Some particular cases are also provided. Among others, the celebrated Kittaneh inequality reads:

w (T) \leq \frac{1}{2} ∥|T| + |T^{*}|∥

. It is [...] Read more.

w (T) \leq \frac{1}{2} ∥|T| + |T^{*}|∥

. It is proved that

w (T) \leq \frac{1}{2} ∥|T| + |T^{*}|∥ - \frac{1}{2} inf_{∥x∥ = 1} {({⟨|T| x, x⟩}^{\frac{1}{2}} - {⟨|T^{*}| x, x⟩}^{\frac{1}{2}})}^{2}

, which improves on the Kittaneh inequality for symmetric and non-symmetric Hilbert space operators. Other related improvements to well-known inequalities in literature are also provided. Full article

(This article belongs to the Special Issue Inequality and Symmetry in Mathematical Analysis)

11 pages, 268 KiB

Open AccessArticle

Generalization of the Lieb–Thirring–Araki Inequality and Its Applications

by Yonggang Li, Jing Wang and Huafei Sun

Mathematics 2021, 9(7), 723; https://doi.org/10.3390/math9070723 - 26 Mar 2021

Viewed by 2173

Abstract

The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, according to the properties of exterior algebra and the Schur-convex function, we provide a new proof for the generalization of the Lieb–Thirring–Araki theorem and Furuta theorem. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

15 pages, 2279 KiB

Open AccessArticle

Robust Control Design to the Furuta System under Time Delay Measurement Feedback and Exogenous-Based Perturbation

by Gisela Pujol-Vazquez, Saleh Mobayen and Leonardo Acho

Mathematics 2020, 8(12), 2131; https://doi.org/10.3390/math8122131 - 29 Nov 2020

Cited by 19 | Viewed by 3121

Abstract

When dealing with real control experimentation, the designer has to take into account several uncertainties, such as: time variation of the system parameters, exogenous perturbation and the presence of time delay in the feedback line. In the later case, this time delay behaviour [...] Read more.

H_{\infty}

theory is presented by employing the linear matrix inequalities techniques to design an efficient output feedback control. This approach is carefully tuned to face with random time-varying measurement feedback and applied to the Furuta pendulum subject to an exogenous ground perturbation. Therefore, a recent experimental platform is described. Here, the ground perturbation is realised using an Hexapod robotic system. According to experimental data, the proposed control approach is robust and the control objective is completely satisfied. Full article

(This article belongs to the Section E2: Control Theory and Mechanics)

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