ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXIV, 1 (2005)
p. 59 - 70

Some change of variable formulas in Integral representation theory
L. Meziani

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 $X=\mathbb{R}$ or $\mathbb{C}$. Now, consider a bounded linear operator $T:C(S,X)\rightarrow Y$ and assume that, due to the effect of a change of variable performed by a bounded operator $V:C(S,X)\rightarrow C(S)$, the operator $T$ takes the product form $T=\theta \cdot V$, with $\theta :C(S)\rightarrow Y$ 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.

Keywords: Change of variable in bounded operators, vector.

AMS Subject classification:  Primary: 28C05, secondary: 46G10.