# S-Spectrum and the quaternionic Cayley transform of an operator

## TL;DR

This paper introduces the quaternionic Cayley transform for symmetric operators in quaternionic Hilbert spaces, exploring its relation to the S-spectrum and defect numbers, advancing the understanding of quaternionic operator theory.

## Contribution

It develops a general theory of the quaternionic Cayley transform and links defect numbers with the S-spectrum, providing new insights into quaternionic operator analysis.

## Key findings

- Established the quaternionic Cayley transform for symmetric operators.
- Analyzed the relationship between defect numbers and the S-spectrum.
- Explored properties of the Cayley transform in quaternionic Hilbert spaces.

## Abstract

In this paper we define the quaternionic Cayley transformation of a densely defined, symmetric, quaternionic right linear operator and formulate a general theory of defect number in a right quaternionic Hilbert space. This study investigates the relation between the defect number and S-spectrum, and the properties of the Cayley transform in the quaternionic setting.