Berberian extension and its S-spectra in a quaternionic Hilbert space

TL;DR

This paper extends the concept of Berberian extension to quaternionic Hilbert spaces, demonstrating its effect on the S-spectrum and applying it to analyze properties of operator commutators.

Contribution

It introduces the Berberian extension in quaternionic Hilbert spaces and shows its role in transforming the S-spectrum, paralleling the complex case.

Findings

Berberian extension converts approximate point S-spectrum into point S-spectrum in quaternionic spaces.

The extension helps analyze properties of commutators of quaternionic operators.

The approach generalizes complex spectral results to quaternionic settings.

Abstract

For a bounded right linear operators $A$, in a right quaternionic Hilbert space $V_{\mathbb{H}}^{R}$, following the complex formalism, we study the Berberian extension $A^\circ$, which is an extension of $A$ in a right quaternionic Hilbert space obtained from $V_{\mathbb{H}}^{R}$. In the complex setting, the important feature of the Berberian extension is that it converts approximate point spectrum of $A$ into point spectrum of $A^\circ$. We show that the same is true for the quaternionic S-spectrum. As in the complex case, we use the Berberian extension to study some properties of the commutator of two quaternionic bounded right linear operators.