On some applications of the ordinary and extended Duhamel products

Let $C_A^{(n)}(D)$ denote the algebra of all $n$-times continuously differentiable functions on $\widebar{D}$ holomorphic on the unit disk $D=\{z\in\mathbf{C}:|z|<1\}$. We prove that $C_A^{(n)}(D)$ is a Banach algebra with multiplication the Duhamel product $(f\circledast g)(z)=\frac{d}{dz}\int_0^zf(z-t)g(t)\,dt$ and describe its maximal ideal space. Using the Duhamel product we prove that the extended spectrum of the integration operator $\mathscr{J}$, $(\mathscr{J}f)(z)=\int_0^zf(t)\,dt$, on $C_A^{(n)}(D)$ is $\mathbf{C}\setminus\{0\}$. We also use the Duhamel product in calculating the spectral multiplicity of a direct sum of the form $\mathscr{J}\oplus A$. We also consider the extension of the Duhamel product and describe all invariant subspaces of some weighted shift operators.