# A fibered description of the vector-valued spectrum

## TL;DR

This paper investigates the structure of the vector-valued spectrum of bounded holomorphic functions between Banach spaces, focusing on fiber descriptions, analytic balls, and Gleason parts to understand its complex geometric properties.

## Contribution

It provides a fibered description of the vector-valued spectrum, exploring the analytic and geometric structure of fibers in the context of Banach space holomorphic functions.

## Key findings

- Characterization of fibers via analytic balls
- Identification of Gleason parts within the spectrum
- Description of the spectrum's geometric structure

## Abstract

For Banach spaces $X$ and $Y$ we study the vector-valued spectrum $\mathcal M_\infty(B_X,B_Y)$, that is the set of non null algebra homomorphisms from $\mathcal H^\infty(B_X)$ to $\mathcal H^\infty(B_Y)$, which is naturally projected onto the closed unit ball of $\mathcal H^\infty(B_Y, X^{**})$. The aim of this article is to describe the fibers defined by this projection, searching for analytic balls and considering Gleason parts.