Spectrum of banach valued holomorphic functions and polar sets

Abstract

Let $\mathcal{A}$ be a complex unital Banach algebra and D be a non-empty open connected subset of $\mathbb{C}.$ Let f be a holomorphic $\mathcal{A}$-valued function on D. We set: $\mathrm{\Sigma }(f)=\{z\in D\mid f(z)\notin Inv(\mathcal{A})\},$where $Inv(\mathcal{A})$ is the set of all invertible elements in $\mathcal{A}$. In this paper, we give a description of $\mathrm{\Sigma }(f)$ using the classical spectrum of $f(z)$ $(z\in D)$. Moreover, within the framework of this paper, we give a partial positive answer to the following problem which was posed by B. Aupetit: If the usual spectrum $\sigma (f(z))$ is polar for all $z\in D,$ is it true that $\mathrm{\Sigma }(f)$ is polar?

Keywords:

• Spectrum
• Banach algebra
• banach-valued holomorphic function
• analytic multivalued function
• pluripolar and polar sets

2020 Mathematics Subject Classifications:

• 47A10
• 47A56

Acknowledgments

The authors would like to express their sincere gratitude to the referee for his careful review of this paper, valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).