Abstract
Let C-A((n))(D) denote the algebra of all n-times continuously differentiable functions on D holomorphic on the unit disk D = {z is an element of C : vertical bar z vertical bar < 1} We prove that C-A((n))(D) is a Banach algebra with multiplication the Duhamel product (f circle star g)(z) (d)/(dz) integral(z)(0) f(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 J, (J f)(z) = integral(z)(0) f(t) dt, on C-A((n))(D) is C\{0}. We also use the Duhamel product in calculating the spectral multiplicity of a direct sum of the form J circle plus A. We also consider the extension of the Duhamel product and describe all invariant subspaces of some weighted shift operators.