Certain Properties Of Essential Numerical Ranges Of Bounded Operators On Banach Spaces

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.

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