On the numerical ranges and Spectra of normal operators

Abstract

The study of numerical ranges and spectra has been of great interest to many mathematicians in the past decades. In this study, we have continued to look at the numerical ranges and spectra of operators .on a Hilbert space. The properties of numerical range, for example, convexity and closedness are well known as proved in the classic Toeplitz - Hausdorff Theorem. In this study, we investigate the relationship between the spectrum and the numerical range of an operator, in particular, when the operator is normal. We have established that for a bounded linear operator on a Hilbert space, the spectrum is contained in the closure of its numerical range. For a normal operator, we have also established that the numerical radius and the spectral radius coincides with the norm of the operator. These results are actually a contribution to the field of numerical ranges and spectra. For us to achieve these, it was paramount that we had a deep understanding of the theory of operators, especially on Hilbert spaces, General Topology, Functional Analysis and Abstract Algebra. This was achieved by reading the available and relevant literature, solving the existing problems and understanding examples in these areas. Further, we also had consultative meetings with the supervisors. In addition, we explored internet Information and further references through the use of research papers in this field. Lastly we could not avoid consultations with other mathematicians who have carried research in this field of study.