Essential numerical range

The convexity, closure and compactness of the numerical range among other properties constitute a considerable literature in operator theory.The properties of the essential numerical range and how they are related to the familiar numerical range are studied. The study underpins the role in operator theory. An outline of the basic concepts and defined terms are provided. Similarly, proofs for simple propositions and theorems used in the sequel are made in Chapter One. Chapter Two of this work is devoted to the properties of the numerical range. Properties, for instance, convexity, unitary invariance and the projection property have been looked at. The connection between the spectrum of an operator and the numerical range of the operator has also been given. The properties of the essential numerical range have been examined in Chapter Three.The proofs of various properties studied, for instance, convexity have been outlined. In Chapter Four, the relationships between the numerical range and the essential numerical range have been studied.