Equivalent Characterizations of Isometrically Bounded Operators with Circular Numerical Ranges

• Author(s): Philip Wafula Mulongo ; Shem Aywa ; John Sirengo
• Paper ID: 1706065
• Page: 315-321
• Published Date: 24-07-2024
• Published In: Iconic Research And Engineering Journals
• Publisher: IRE Journals
• e-ISSN: 2456-8880
• Volume/Issue: Volume 8 Issue 1 July-2024

Abstract

This study investigates the equivalent characterizations of isometrically bounded operators with circular numerical ranges on complex Hilbert spaces. The main objective is to establish necessary and sufficient conditions for an isometrically bounded operator to have a circular numerical range and to explore the connections between circularity and unitary equivalence, scalar multiples of unitary operators, and scalar multiples of isometries. The study employs a comprehensive theoretical framework that combines techniques from functional analysis, operator theory, and convex geometry to derive the main results. The key findings include a complete characterization of isometrically bounded operators with circular numerical ranges in terms of their unitary equivalence and representation as scalar multiples of unitary operators or isometries. The study also presents several related results on the spectral properties and geometric structure of these operators. Furthermore, the research highlights the potential applications of the findings in quantum mechanics and matrix analysis, where the circularity of numerical ranges plays a crucial role. The study makes significant contributions to the understanding of isometrically bounded operators and their numerical ranges, providing a unified and generalized framework for analyzing their properties and behavior. The results and techniques developed in this research have the potential to inspire new approaches to problems in operator theory and related fields, and to lead to the development of novel algorithms and tools for studying the geometry of numerical ranges.

Keywords

Equivalent Characterizations, Isometrically Bounded Operators, Circular Numerical Ranges

Citations

IRE Journals:
Philip Wafula Mulongo , Shem Aywa , John Sirengo "Equivalent Characterizations of Isometrically Bounded Operators with Circular Numerical Ranges" Iconic Research And Engineering Journals Volume 8 Issue 1 2024 Page 315-321

IEEE:
Philip Wafula Mulongo , Shem Aywa , John Sirengo "Equivalent Characterizations of Isometrically Bounded Operators with Circular Numerical Ranges" Iconic Research And Engineering Journals, 8(1)