The theme of this dissertation is the weak compactness of the unit sphere in algebr,as of operators on Banach spaces. These algebras (in Chapters 2 and 3) are the images of c(Lambda) (I. 1) and A4, (III. 2) under an operational calculus for an operator on a Banach space. We show (Chapter 2) that the it condition of relative weak compactness of the Unit sphere characterises the scalar-type spectral operators among those with a. C(Lambda)-operational calculus.