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 R H , following the complex formalism, we study the Berbe rian extension A◦, which is an extension of A in a right quaternionic Hilbert space obtained from V R H . 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◦. 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.