| dc.description.abstract | Abstract
The numerical range of an operator on a Hilbert space has been extensively
researched on. The concept of numerical range of an operator goes back
as early as 1918 when Toeplitz defined it as the field of values of a matrix
for bounded linear operators on a Hilbert space. Major results like convexity,
that is the Toeplitz-Hausdorff theorem, the relationship of the spectrum
and the numerical range, the essential spectra and the essential numerical
range, have given a lot of insights. Most of these results have been on Hilbert
spaces. As for Banach spaces there is still work to be done. There is scanty
literature on the properties of the essential spectra and the essential numerical
ranges on Banach spaces. The objectives of this study were to determine
the properties of the essential spectrum and the properties of the essential
numerical range, and to investigate the relationship between the essential
spectrum and the essential numerical range for operators on Banach spaces.
To study the properties of the essential spectra, we defined various parts of
the spectra and using known theorems, we established the duality properties
of these parts. For the essential numerical range, we applied the approach
of Barraa and M¨ uller which considers a measure of noncompactness instead
of the usual essential norm on the Calkin algebra. We finally extended the
existing relations between the spectra and the numerical range to the setting
of the essential spectrum and essential numerical range. We hope that the
results of this study will be significant to both Applied Mathematicians and
theoretical physicists for further research. | en_US |
