Spatial Numerical range of bounded operators on right quaternionic Banach spaces

In this paper, we establish and study some properties of the spatial numerical range of right linear bounded operators on a right quaternionic Banach space. To be more specific, we show that the spatial numerical range is circular and we give the relation between the spatial numerical range, the point S-spectrum and the approximate S-spectrum of an operator on a right quaternionic Banach space. We prove also that the S-spectrum of a quaternionic bounded operator is included in the closure of its spatial numerical range. To show this, we generalize the Bishop-Phelps theorem for quaternionic Banach spaces.