On the relation between S-spectrum and right spectrum

We use the $\mathbb{R}$-linearity of $I\lambda -T$ to define $\sigma (T)$, the right spectrum of a right $\mathbb{H}$-linear operator $T$ in a right quaternionic Hilbert space. We show that $\sigma (T)$ coincides with the $S$-spectrum ${\sigma }_S(T)$.