| dc.description.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. | en_US |
