On the Banach algebra structure for C(n) of the bidisc and related topics

Abstract

Let C(n)=C(n)(D×D) be a Banach space of complex valued functions f(x,y) that are continuous on the closed bidisc D×D¯, where D={z∈C:|z|<1} is the unit disc in the complex plane C and has nth partial derivatives in D×D which can be extended to functions continuous on D×D¯. The Duhamel product is defined on C(n) by the formula (f⊛g)(z,w)=∂2∂z∂w∫0z∫0wf(z−u,w−v)g(u,v)dvdu. In the present paper we prove that C(n)(D×D) is a Banach algebra with respect to the Duhamel product ⊛. This result extends some known results. We also investigate the structure of the set of all extended eigenvalues and extended eigenvectors of some double integration operator Wzw. In particular, the commutant of the double integration operator Wzw is also described.