Berberian Extension and its S-spectra in a Quaternionic Hilbert Space

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.