**
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.

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