Current Volume 8
Abstract
This paper investigates the circularity of numerical ranges for isometrically bounded operators on Hilbert spaces. We establish a complete characterization of the class of isometrically bounded operators with circular numerical ranges in terms of their unitary equivalence to scalar multiples of unitary operators, which we call the Circular Numerical Range Theorem. Several corollaries and equivalent formulations of this result are derived, unifying and extending previous results on the circularity of numerical ranges for specific operator classes. Furthermore, we establish the Circular Boundary Theorem, relating the circularity of the numerical range to the essential spectrum of the operator, and the Circular Convex Hull Theorem, characterizing circular numerical ranges as the closed convex hull of the essential spectrum. These results offer new insights into the relationship between the geometry of the numerical range and the spectral properties of the operator. Throughout the paper, we provide carefully chosen examples and counterexamples to illustrate the key aspects of the theory and demonstrate the sharpness of our findings. The implications of our results for operator theory and potential applications in various fields, such as quantum mechanics and matrix analysis, are discussed. Our work contributes to the general theory of numerical ranges and opens up new avenues for research in functional analysis and related areas.
Keywords
Circularity, Numerical, Isometrically
Citations
IRE Journals:
Joel S Barasa , Lucy Chikamai , Shem Aywa
"Circularity of Numerical Ranges for Isometrically Bounded Operators" Iconic Research And Engineering Journals Volume 7 Issue 12 2024 Page 144-150
IEEE:
Joel S Barasa , Lucy Chikamai , Shem Aywa
"Circularity of Numerical Ranges for Isometrically Bounded Operators" Iconic Research And Engineering Journals, 7(12)
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