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Keywords = Selberg operator

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17 pages, 333 KiB  
Article
Norm and Numerical Radius Inequalities Related to the Selberg Operator
by Najla Altwaijry, Cristian Conde, Silvestru Sever Dragomir and Kais Feki
Symmetry 2023, 15(10), 1860; https://doi.org/10.3390/sym15101860 - 3 Oct 2023
Cited by 2 | Viewed by 1156
Abstract
The main focus of this paper is the study of the Selberg operator. It aims to establish appropriate bounds for the norm and numerical radius of the product of three bounded operators, with one of them being a Selberg operator. Moreover, it offers [...] Read more.
The main focus of this paper is the study of the Selberg operator. It aims to establish appropriate bounds for the norm and numerical radius of the product of three bounded operators, with one of them being a Selberg operator. Moreover, it offers several bounds involving the summation of operators, notably the Selberg operator. Through the examination of these properties and relationships, this study contributes to a better understanding of the Selberg operator and its influence on operator compositions. The paper also highlights the significance of symmetry in mathematics and its potential implications across various mathematical domains. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Functional Equations)
15 pages, 317 KiB  
Article
Some Refinements of Selberg Inequality and Related Results
by Najla Altwaijry, Cristian Conde, Silvestru Sever Dragomir and Kais Feki
Symmetry 2023, 15(8), 1486; https://doi.org/10.3390/sym15081486 - 27 Jul 2023
Cited by 5 | Viewed by 1533
Abstract
This paper introduces several refinements of the classical Selberg inequality, which is considered a significant result in the study of the spectral theory of symmetric spaces, a central topic in the field of symmetry studies. By utilizing the contraction property of the Selberg [...] Read more.
This paper introduces several refinements of the classical Selberg inequality, which is considered a significant result in the study of the spectral theory of symmetric spaces, a central topic in the field of symmetry studies. By utilizing the contraction property of the Selberg operator, we derive improved versions of the classical Selberg inequality. Additionally, we demonstrate the interdependence among well-known inequalities such as Cauchy–Schwarz, Bessel, and the Selberg inequality, revealing that these inequalities can be deduced from one another. This study showcases the enhancements made to the classical Selberg inequality and establishes the interconnectedness of various mathematical inequalities. Full article
(This article belongs to the Special Issue Symmetry in Functional Analysis and Operator Theory)
16 pages, 342 KiB  
Article
On the Generalised Transfer Operators of the Farey Map with Complex Temperature
by Claudio Bonanno
Mathematics 2023, 11(1), 134; https://doi.org/10.3390/math11010134 - 27 Dec 2022
Viewed by 1456
Abstract
We consider the problem of showing that 1 is an eigenvalue for a family of generalised transfer operators of the Farey map. This is an important problem in the thermodynamic formalism approach to dynamical systems, which in this particular case is related to [...] Read more.
We consider the problem of showing that 1 is an eigenvalue for a family of generalised transfer operators of the Farey map. This is an important problem in the thermodynamic formalism approach to dynamical systems, which in this particular case is related to the spectral theory of the modular surface via the Selberg Zeta function and the theory of dynamical zeta functions of maps. After briefly recalling these connections, we show that the problem can be formulated for operators on an appropriate Hilbert space and translated into a linear algebra problem for infinite matrices. This formulation gives a new way to study numerically the spectrum of the Laplace–Beltrami operator and the properties of the Selberg Zeta function for the modular surface. Full article
(This article belongs to the Special Issue Advances in Ergodic Theory and Its Applications)
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