Some applications of Banach algebra techniques

We consider some functional Banach algebras with multiplications as the usual convolution product * and the so-called Duhamel product circle star. We study the structure of generators of the Banach algebras (C-(n) [0, 1], *) and (C-(n) [0, 1], circle star). We also use the Banach algebra techniques in the calculation of spectral multiplicities and extended eigenvectors of some operators. Moreover, we give in terms of extended eigenvectors a new characterization of a special class of composition operators acting in the Lebesgue space L-p [0, 1] by the formula (C-phi f) (x) = f (phi (x)). (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim