On Some Applications of the Ordinary and Extended Duhamel Products

Abstract

Let C A () (D) denote the algebra of all n-times continuously differentiable functions on $$\overline D$$ holomorphic on the unit disk D = {z ∈ C : |z| < 1}. We prove that C A () (D) is a Banach algebra with multiplication the Duhamel product $$(f \circledast g)(z) = \tfrac{d}{{dz}}\smallint _0^z f(z - t)g(t)dt$$ and describe its maximal ideal space. Using the Duhamel product we prove that the extended spectrum of the integration operator $$J,\;(Jf)(z) = \smallint _0^z f(t)\;dt$$ , on C A () (D) is C\{0}. We also use the Duhamel product in calculating the spectral multiplicity of a direct sum of the form ℐ ⊕ A. We also consider the extension of the Duhamel product and describe all invariant subspaces of some weighted shift operators.