Illinois Journal of Mathematics

On the Banach algebra structure for $C^{(\mathbf{n})}$ of the bidisc and related topics

Ramiz Tapdıgoglu

Abstract

Let $C^{(\mathbf{n})}=C^{(\mathbf{n})}(\mathbb{D}\times \mathbb{D})$ 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 $\mathbb{D}=\{z\in \mathbb{C}:|z|\lt 1\}$ 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 $C^{(\mathbf{n})}$ by the formula $\begin{equation*}(f\circledast g)(z,w)=\frac{\partial ^{2}}{\partial z\partial w}\int _{0}^{z}\int _{0}^{w}f(z-u,w-v)g(u,v)\,dv\,du.\end{equation*}$ In the present paper we prove that $C^{(\mathbf{n})}(\mathbb{D}\times\mathbb{D})$ is a Banach algebra with respect to the Duhamel product $\circledast$. 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.

Article information

Source
Illinois J. Math., Volume 64, Number 2 (2020), 185-197.

Dates
Revised: 5 December 2019
First available in Project Euclid: 1 May 2020

https://projecteuclid.org/euclid.ijm/1588298627

Digital Object Identifier
doi:10.1215/00192082-8303477

Mathematical Reviews number (MathSciNet)
MR4092955

Zentralblatt MATH identifier
07210956

Citation

Tapdıgoglu, Ramiz. On the Banach algebra structure for $C^{(\mathbf{n})}$ of the bidisc and related topics. Illinois J. Math. 64 (2020), no. 2, 185--197. doi:10.1215/00192082-8303477. https://projecteuclid.org/euclid.ijm/1588298627

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