Some Problems of Spectral Theory

Also compact operators of scalar type in any Banach can be so expressed, provided, the sum (1) is allowed to be countably infinite. It is perhaps less often noted that any operator of scalar type can be expressed in the form (1), with the index j running over all positive integers. But, in general, the projections P. cannot be chosen pair-wise J disjoint, that is, the product of any distinct pair of them might not be equal to the zero-operator. In fact, an operator T in a space E is of scalar type if and only if, there exists an abstract space ~' a a-algebra S of its subsets, a cr-additive and multiplicative measure P S -+ L(E) such that P(~) = I, the identity operator, and a P-integrable function f such that