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.

15 pages. arXiv admin note: text overlap with arXiv:1805.01461, arXiv:1905.05936, arXiv:1904.02977, arXiv:1805.10131, arXiv:1705.05240

References in corpus (1)

• Kato S-spectrum in the quaternionic setting