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

Abstract

Let $C^{\left(\mathbf{n}\right)}=C^{\left(\mathbf{n}\right)}(\mathbb{D}\times \mathbb{D})$ be a Banach space of complex valued functions $f(x,y)$ that are continuous on the closed bidisc $\overline{\mathbb{D}\times \mathbb{D}}$, where $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ is the unit disc in the complex plane $\mathbb{C}$ and has $n$th partial derivatives in $\mathbb{D}\times \mathbb{D}$ which can be extended to functions continuous on $\overline{\mathbb{D}\times \mathbb{D}}$. The Duhamel product is defined on $C^{\left(\mathbf{n}\right)}$ by the formula $$\left(f⊛g\right)(z,w)=\frac{{\partial }^2}{\partial z\partial w}{\int }_0^z{\int }_0^{w}f(z-u,w-v)g(u,v)dvdu.$$ In the present paper we prove that $C^{\left(\mathbf{n}\right)}(\mathbb{D}\times \mathbb{D})$ 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 $W_{zw}$. In particular, the commutant of the double integration operator $W_{zw}$ is also described.