The theme of this thesis is the construction of state diagrams and their implications. The author generalises most of the theorems in Chapter II of Goldberg [Gl] by dropping the assumption that the doin.ain of the operator is dense in X . The author also presents the standard Taylor-Halberg-Goldberg state diagrams [Gl, 61, 66]. Chapters II and III deal with F₊- and F₋-operators, which are generalisations of the ф₊- and ф₋-operators in Banach spaces of Gokhberg-Krein [GK]. Examples are given of F₊- and F₋-operators. Also, in Chapter III, the main theorems needed to construct the state diagrams of Chapter IV are discussed. The state diagrams of Chapter IV are based on states corresponding to F₊- and F₋-operators; in addition state diagrams relating T and T˝ under the assumptions ϒ(T) > 0 and ϒ(T΄) > 0 are derived. Second adjoints are important in Tauberian Theory (see Cross [Cl]). Chapters I and IV are the main chapters. In Chapter I of this thesis the author modifies many of the proofs appearing in Goldberg [Gl), to take account of the new definition of the adjoint.

Bibliography: pages 107-110.