Algebras of compact operators

The purpose of this thesis is to examine certain classes of hounded linear operators on a Banach space X in an algebraic light, i.e. as elements of a Banach algebra rather than as operators on X, the Banach algebra in general being the algebra B(X) of all bounded linear operators on X. We choose those properties which can be expressed in general algebraic terms, and then study elements of a general Banach algebra which satisfy these properties. The class originally chosen, suggested to me by Professor P. P. Bonsall, was the class of compact operators on X. As the algebraic properties of such operators generally involve their spectral properties, it was natural to extend our study to include Riesz operators as well