About the Cauchy–Bunyakovsky–Schwarz Inequality for Hilbert Space Operators

The main focus of this paper is on establishing inequalities for the norm and numerical radius of various operators applied to a power series with the complex coefficients h(λ)=∑k=0∞akλk and its modified version ha(λ)=∑k=0∞|ak|λk. The convergence of h(λ) is assumed on the open disk D(0,R), where R is the radius of convergence. Additionally, we explore some operator inequalities related to these concepts. The findings contribute to our understanding of operator behavior in bounded operator spaces and offer insights into norm and numerical radius inequalities.

Section: Introduction and Preliminarymentioning

confidence: 99%

Norm and Numerical Radius Inequalities for Sums of Power Series of Operators in Hilbert Spaces

Altwaijry,

Dragomir,

Feki

2024

“…Inequalities play a crucial role in analysis and find applications in various areas of mathematics (see [1][2][3][4][5][6][7] and related sources). Among these, Bessel's inequality and the Boas-Bellman inequality hold significant importance and are widely used in the study of operators on Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

New Results on Boas–Bellman-Type Inequalities in Semi-Hilbert Spaces with Applications

Altwaijry

Dragomir

Feki

2023

In this article, we investigate new findings on Boas–Bellman-type inequalities in semi-Hilbert spaces. These spaces are generated by semi-inner products induced by positive and positive semidefinite operators. Our objective is to reveal significant properties of such spaces and apply these results to the field of multivariable operator theory. Specifically, we derive new inequalities that relate to the joint A-numerical radius, the joint operator A-seminorm, and the Euclidean A-seminorm of tuples of semi-Hilbert space operators. We assume that A is a nonzero positive operator. Our discoveries provide insights into the structure of semi-Hilbert spaces and have implications for a broad range of mathematical applications and beyond.

Some New Estimates for the Berezin Number of Hilbert Space Operators

Altwaijry

Feki

Minculete

2022

In this paper, we have developed new estimates of some estimates involving the Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space HΩ. The uniqueness or novelty of this article consists of new estimates of Berezin numbers for different types of operators. These estimates improve the upper bounds of the Berezin numbers obtained by other similar papers. We give several upper bounds for berr(S*T), where T,S∈B(HΩ) and r≥1. We also present an estimation of ber2r∑i=1dTi where Ti∈B(HΩ), i=1,d¯ and r≥1. Some of the obtained inequalities represent improvements to earlier ones. In this work, the ideas and methodologies presented may serve as a starting point for future investigation in this field.