Simultaneous zero inclusion property for spatial numerical ranges

For a finite dimensional complex normed space X, we say that it has the simultaneous zero inclusion property if an invertible linear operator S on X has zero in its spatial numerical range if and only if zero is in the spatial numerical range of the inverse S−1, as well. We show that beside Hilbert spaces there are some other normed spaces with this property. On the other hand, space ℓ1(n) does not have this property. Since not every normed space has the simultaneous zero inclusion property, we explore the class of invertible operators at which this property holds. In the end, we consider a property which is stronger than the simultaneous zero inclusion property and is related to the question when it is possible, for every invertible operator S, to control the distance of 0 to the spatial numerical range of S−1 by the distance of 0 to the spatial numerical range of S.