Numerical range of quaternionic right linear bounded operators

Abstract

In this paper, we prove that for a right linear bounded operator on a quaternionic Hilbert space, the norm and the numerical radius are equal if and only if the norm and the spectral radius are equal. We also show that the spherical spectrum of a quaternionic bounded operator is included in the closure of its numerical range, and we show that the numerical range of an operator on a quaternionic Hilbert space is not necessarily convex. For a quaternionic bounded normal operator, we prove that the convex hull of the closure of its numerical range is equal to the convex hull of its spherical spectrum. Finally, we give some inequalities between the numerical radius, the spectral radius and the norm of a right linear bounded operator, and we prove also that the norm and the numerical radius of a quaternionic bounded hyponormal operator are equal.

Keywords:

• Quaternionic Hilbert space
• normal operator
• numerical range
• numerical radius

2010 Mathematics Subject Classifications:

• 47B15
• 47S10
• 35P05

Acknowledgements

The authors thank the referee for his useful comments that helped to improve the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).