Some applications of the Duhamel product (Q937889)

The author uses the Duhamel product defined by the formula \[ f\otimes g (x)= \frac{d}{dx}\int_0^x f(x-t)g(t) \,dt \] defined for functions in \(C^1([0,1])\) and the corresponding formula \[ f\otimes g (x,y)= \frac{\partial}{\partial x \partial y}\int_0^x\int_0^y f(x-t,y-s)g(t,s) \,dt\,ds \] defined for functions in \(C^{(n)}([0,1])\times [0,1])\) for \(n\geq 2\) to show that on the subspace of functions depending on the product \(xy\) the cyclic vectors of the Volterra integration operator in two variables \(Wf(xy)=\int_0^x\int_0^y f(ts)\,dt\,ds\) coincide with the functions such that \(f(0)\neq 0\). The paper also contains general information on the Duhamel product and its properties when defined on analytic functions in the unit disc. A theorem concerning its use to calculate multiplicities of spectra for direct sums \(J\otimes A\) where \(J\) is the integration operator acting on the Wiener algebra and \(A\) is an operator on a separable Banach space with \(\| A^n\| \leq \frac{c}{n!}\) is also obtained.