Some applications of Duhamel product

The Duhamel product of functions $f$ and $g$ is defined by formula
$$ (f\circledast g)(x)=\frac{d}{dx}\int^x_0 f(x-t)g(t)\,dt. $$
In the present paper the Duhamel product is used in the study of the spectral multiplicity for direct sums of operators and in the description of cyclic vectors of the restriction of the integration operator in two variables $f(x,y)\mapsto\int^x_0\int^y_0 f(t,\tau)d\tau\,dt$ to its invariant subspace consisting of functions that depend only on the product $xy$.