# Decomposable operators, local S-spectrum and S-spectrum in the quaternionic setting

## TL;DR

This paper explores the properties of decomposable operators and their spectra in quaternionic Hilbert spaces, establishing connections between various spectral concepts and analyzing local S-spectra using slice-regular functions.

## Contribution

It introduces a quaternionic framework for decomposable operators and studies the relationships between different S-spectra, extending spectral theory in quaternionic Hilbert spaces.

## Key findings

- S-spectrum, approximate S-point spectrum, surjectivity S-spectrum, and local S-spectra coincide for decomposable operators
- Properties of local S-spectrum and local S-spectral subspaces are characterized using slice-regular functions
- Connections between Bishop's property, single valued extension property, and decomposability are established

## Abstract

In a right quaternionic Hilbert space, following the complex formalism, decomposable operators, the so-called Bishop's property and the single valued extension property are defined and the connections between them are studied to certain extent. In particular, for a decomposable operator, it is shown that the S-spectrum, approximate S-point spectrum, surjectivity S-spectrum and the union of all local S-spectra coincide. Using continuous right slice-regular functions we have also studied certain properties of local S-spectrum and local S-spectral subspaces.