Maximal ideal space of some Banach algebras and related problems

Abstract

Let $C_{A}^{\left( n\right) }:=C_{A}^{\left( n\right) }\left( \mathbb{D}\times\mathbb{D}\right) $ denote the subspace of functions in the Banach space $C^{\left( n\right) }\left( \overline{\mathbb{D}% \times\mathbb{D}}\right) $ which are analytic in the bi-disc $\mathbb{D}% \times\mathbb{D}$. We consider the subspace $B_{zw}$ consisting from the functions $f\in C_{A}^{\left( n\right) }$ which can be represented in the form $f\left( z,w\right) =g\left( zw\right) ,$ where $g$ is a single variable function from the disc algebra $C_{A}\left( \mathbb{D}\right) $. We prove that $B_{zw}$ is a Banach algebra under the Duhamel multiplication \[ \left( f\circledast g\right) \left( zw\right) =\frac{\partial^{2}% }{\partial z\partial w}\underset{0}{\overset{z}{\int}}\underset{0}{\overset {w}{\int}}f\left( \left( z-u\right) \left( w-v\right) \right) g\left( uv\right) dvdu \] and describe its maximal ideal space. We also consider the Hardy type operator $f\rightarrow xy\underset{0}{\overset{x}{\int}}\underset{0}{\overset{y}{\int}% }f\left( t\tau\right) d\tau dt$ and discuss its some properties.