Abstract
Let X, Y be Banach spaces and let us denote by C(S, X) the space of all X-valued continuous functions on the compact Hausdorff space S, equipped with the uniform norm. We shall write C(S, X) = C(S) if X = R or C. Now, consider a bounded linear operator T : C(S, X) -> Y and assume that, due to the effect of a change of variable performed by a bounded operator V : C(S, X) -> C(S), the operator T takes the product form T = theta.V, with theta : C(S) -> Y linear and bounded. In this paper, we prove some integral formulas giving the representing measure of the operator T, which appeared as an essential object in integral representation theory. This is made by means of the representing measure of the operator theta which is generally easier. Essentially the estimations are of the Radon-Nikodym type and precise formulas are stated for weakly compact and nuclear operators.