On Some Applications of the Ordinary and Extended Duhamel Products

Author： Karaev M.  

Publisher： Springer Publishing Company

ISSN： 0037-4466

Source： Siberian Mathematical Journal, Vol.46, Iss.3, 2005-05, pp. : 431-442

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Abstract

Let C A(n) (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(n) (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(n) (D) is C{0}. We also use the Duhamel product in calculating the spectral multiplicity of a direct sum of the form ⊕ A. We also consider the extension of the Duhamel product and describe all invariant subspaces of some weighted shift operators.

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